Quantitative functional calculus in Sobolev spaces

In the framework of Sobolev (Bessel potential) spaces $H^n(\reali^d, \reali {or} \complessi)$, we consider the nonlinear Nemytskij operator sending a function $x \in \reali^d \mapsto f(x)$ into a composite function $x \in \reali^d \mapsto G(f(x), x)$. Assuming sufficient smoothness for $G$, we give a"tame"bound on the $H^n$ norm of this composite function in terms of a linear function of the $H^n$ norm of $f$, with a coefficient depending on $G$ and on the $H^a$ norm of $f$, for all integers $n, a, d$ with $a>d/2$. In comparison with previous results on this subject, our bound is fully explicit, allowing to estimate quantitatively the $H^n$ norm of the function $x \mapsto G(f(x),x)$. When applied to the case $G(f(x), x) = f^2(x)$, this bound agrees with a previous result of ours on the pointwise product of functions in Sobolev spaces.


Introduction.
Functional calculus in Sobolev spaces has been extensively discussed, from the classic papers [1] [4] to the recent book [19]. These and the other works of our knowledge have been mainly devoted to the well definedness of nonlinear composition operators (Nemytskij operators), and to the qualitative structure of the norm estimates for these operators. On the contrary, in this paper we are interested in giving fully quantitative norm estimates for the Nemytskij operators, for the special case of L 2 type Sobolev spaces, including evaluation of all numerical constants involved therein; the spirit is the same as in the previous work [18] on the pointwise product of two functions in a Sobolev space, whose results imply in an elementary way estimates on the operators of composition with polynomials. Here we will infer estimates for much more general Nemytskij operators, covering essentially the composition with any sufficiently derivable mapping. When the present, general results are applied to polynomial mappings, in the quadratic case they reproduce the upper bounds arising from [18] (whose reliability was also tested in the cited work, by comparison with the lower bounds obtained from appropriate trial functions); for polynomial Nemytskij operators of degree ≥ 3, the upper bounds derived in the present framework are even more efficient than the elementary iteration of the estimates in [18]. The Sobolev spaces we consider in this paper are the L 2 type Sobolev (Bessel potential) spaces H n (R d , C), or their real analogues H n (R d , R), with the corresponding norms n := H n (see [2] [13] or Sect.2); the attention is restricted to the case of integer order n. Before describing our results on Nemytskij operators, let us fix some notations. Consider a function G : B × R d → R or C, where B is an interval of R or a ball of C, say, with center 0; if f : R d → B and χ : R d → R d , one can define the composition G(f, χ) : R d → R or C, x → G(f (x), χ(x)). We choose for χ the identity mapping x : R d → R d , x = (x 1 , ..., x d ) → x; thus The (generally nonlinear) mapping will be called the Nemytskij operator associated to G. Its domain is the set of functions (modulo equality almost everywhere) f : R d → B, x → f (x), and the codomain is made of functions R d → R or C (in the sequel, the expression "G(f, x) is defined " will always be employed to mean that a function f on R d takes values in B).
The classical problem of functional calculus in Sobolev spaces is to prove that G(f, x) is in a Sobolev space, and to estimate its norm, when G is sufficiently smooth and f is in a related Sobolev space. To discuss this topic one can fix a base integer a > d/2, so that functions in H a be bounded and continuous (in applications, a is often chosen to be the smallest integer > d/2); a second integer n runs freely over N, labelling the scale of all H n spaces. It is known that if G satisfies suitable conditions of smoothness and boundedness, G(0, x) is in H n and f is in H n ∩ a ball of H a , then G(f, x) is defined, it belongs to H n and as it is seen, the r.h.s. in this inequality is a linear function of f n , with a coefficient depending on f a in a way determined by n, a, d and by the function G.
Our aim is to provide an explicit expression for the function Υ nad ruling this dependence: this will be given in Prop.s 2.4 and 2.7, which are the main results of the paper. One feature of the expression we will obtain is that it does not depend dramatically on the dimension d (differently from the bounds frequently appearing in similar situations, when the indices of derivatives are manipulated too naively); this result is obtained by systematic use of tensorial methods to express the derivatives of any order of functions on R d and their composition with G. Another feature of our result, already stressed, is that it contains, as a subcase, the outcomes of [18] on the case G(f, x) = f 2 = the pointwise product of f by itself (estimates on pointwise product were derived in [18] using the Fourier transform, instead of the present tensor approach, as an alternative tool to deal efficiently with the d-dependence).
In Eq.(1.3) one recognises the structure of the so-called "tame" estimates, forming the basis of the Nash-Moser implicit function theorem [9]. More precisely, we have a "tame estimate of order zero" (an estimate of general order r would give the n-th norm of a nonlinear operator on f in terms of a linear function of f n+r , with a coefficient depending on f a for some fixed order a). Such zero order estimates are used in [16] in relation to semilinear evolution equations on the intersection ∩ ∞ n=0 H n ; the results of the present paper allow quantitative estimates on the existence times for the solutions, when the nonlinear part of the evolution equation is a Nemytskij operator. Let us illustrate the organisation of the paper. In Sect.2 we describe the main results, i.e., the expression of Υ nad in the case of real and complex Sobolev spaces; this requires some preliminaries on the tensor formalism and the definition of some "universal" polynomials P m (m = 1, 2, 3, ...) in terms of generating relations. The rest of the paper is devoted to the proofs. In Sect.3 we write a Leibnitz rule for the derivatives of tensor fields on R d , and present a generalized Faà di Bruno formula for the derivatives of any order of G(f, x); both constructions rely on symmetrised tensor products, which are conveniently discussed. In Sect.4 we prove the main properties of the polynomials P m . In Sect.5 we discuss some functional spaces of tensor fields and the Hausdorff-Young inequality for the Fourier transform of tensors; also, we propose estimates for the constants in a Gagliardo type and an Adams-Frazier type inequality. In Sect.6 we use these tools to derive the expression of Υ nad presented in Sect.2. Finally, in the Appendix we prove some technical statements on tensor norms and the symmetric tensor calculus, which are necessary to derive the main results of the paper.
2 Description of the results. Tensor notations and Sobolev spaces. For our purposes, it is convenient to introduce some tensor representations for expressions with indices, and some related spaces of tensor-valued functions. This setting will simplify the exposition of the results and the subsequent proofs. For m ∈ N 0 , we indicate with ⊗ m C d the tensor product of m copies of C d ; this will be identified with the complex vector space of families where T λ ∈ C for all λ; any such T will be called a complex tensor of order m. The norm of T is In the sequel, we always intend ⊗ 0 C d := C (as customary for tensor products). Let p ∈ [1, +∞]; then, L p (R d , ⊗ m C d ) is the space of measurable functions (modulo equality almost everywhere) such that the function .., d} m . The L p norm of a function T as above is We use the standard notations D(R d , C), D ′ (R d , C) for the space of smooth, compactly supported functions R d → C and the space of distributions. More generally, D(R d , ⊗ m C d ) is the space of smooth, compactly supported functions Φ : where ∂ λ := ∂ λ 1 ...∂ λm (∂ λ i the partial derivative in the λ i -th direction) and the derivatives are intended in the ordinary or in the distributional sense; we also intend ∇ 0 f := f . In the above tensor style, the L 2 type space of complex Bessel potentials of integer order n ∈ N can be defined setting this space carries the (Hilbertian) norm Definitions (2.7) and (2.8) will be our standards throughout the paper. In the integer case we consider, they coincide exactly with the conventional definition of H n [2] [13] in terms of the operator It should be noted that, due to the permutability of partial derivatives, the derivatives ∇ m f of a real or complex function f take values in the space of symmetric tensors, on which we will return later from a more systematic viewpoint.
Useful constants and notations. For any integer a > d/2, we put Γ denoting the factorial function. This is the sharp constant in the imbedding and S ad is the minimum constant for which this occurs [17]. For a as before and r ∈ (0, +∞], we put (intending this to be the whole H a space, if r = +∞); then (all this can be extended to the case of a noninteger a > d/2, but here we are not considering fractional Sobolev spaces).
Using this function, we define for all integers m, j with 1 ≤ j ≤ m. It should be noted that for m, j as above; in fact, U mjd is a product of factors of the form E(s)/E(1 − s), with s ∈ [0, 1/2], raised to power d/2; each one of these factors is ≤ 1. We will show how the constant U mjd arises in an Adams-Frazier type inequality for products of derivatives (see Prop.5.3).
Defining the "universal" polynomials P m . As anticipated in the Introduction, these polynomials will be employed to express the main results on Nemytskij operators; we will give some equivalent descriptions for them via the following Lemma, to be proved in Sect.4.

2.1
Lemma. Let m ∈ N 0 . Then, there is a unique real polynomial P m ((ν jℓ ), ρ), in a set of variables

16)
such that for each C m function F : This polynomial has the form where the P mjℓ are positive, integer coefficients; these have the equivalent characterizations 20) and can be described as the solutions of the recursive equations P 110 = 1 , P 100 = 0 ;  Table. There is a strict relation between the coefficients P mjℓ and the Stirling numbers. Let us recall that the Stirling number of the second kind S j m is defined to be the number of partitions of a set of m elements into j nonempty subsets; comparing some standard results on these numbers [5] with anyone of Eq.s (2.19-2.21), it is found that (In particular, P mj0 = S j m ; these are the only coefficients involved in the derivative (2.17) in the case of a ξ-independent function F (u), which is considered in [5].

2.3
Definition. Let n ∈ N. A map G as above is said to have the Π n property if: i) G is C n ; ii) ♭ m (G, ρ) < +∞ for 0 ≤ m ≤ n and all ρ ∈ (0, r); σ jℓ (G, ρ) < +∞ for 1 ≤ j ≤ n, 0 ≤ ℓ ≤ n − j and all ρ ∈ [0, r); is defined on the grounds of Eq.(2.12). With the above terminology, the main result on real Nemystkij operators is the following.
2.4 Proposition. Let n, a ∈ N and a > d/2. Consider: i) a function G : and where, for any ρ ∈ [0, r), (intending these to be ♭ 0 (G, ρ) and 0, respectively, for n = 0). Eq.(2.31) implies the weaker bound ii) We recall that β md and b md depend on the constants U m−ℓ,jd , all of them ≤ 1; weaker but simpler estimates can be obtained replacing systematically those constants with one.
iii) The previous results become simpler if G is x-independent: G(u, x) = G(u); this case is frequently considered in the investigation of Nemytskij operators and will be discussed in more detail at the end of this Section, also giving two examples. A main feature of this case is that c nd (G, ρ) = 0 for all n, ρ. ⋄ Nemytskij operators in the complex case. From now on B C (0, r) denotes the open ball in C of center 0 and radius r, and B C (0, r) its closure. We consider a function with r ∈ (0, +∞]. For partial derivatives with respect to the complex variable, we use the notations we write ∂ λ , as in Eq.(2.25), for the derivatives w.r.t. the x variables. We say that G is C n if all partial derivatives of G of order ≤ n w.r.t. z, z and (x 1 , ..., x d ) exist and are continuous, everywhere on its domain. We put for h, k, ℓ ∈ N and G of class C h+k+ℓ ; this can be regarded as a function for m, h, k, j, ℓ integers and G of class C m , C h+k+ℓ and C j+ℓ , respectively (these quantities being possibly infinite). For m ≥ 1 integer and ρ ∈ (0, r), the functions β md (G, ρ), b md (G, ρ) will be defined as in Def.2.2, using again the polynomials P m of Lemma 2.1. ⋄ As in the real case, the indicated suprema are monotonically increasing with ρ, and their finiteness at ρ 1 ensures finiteness for all ρ ≤ ρ 1 . If ♭ m (G, ρ) < +∞, at least for ρ in a right neighborhood of zero, we put ♭ m (G, 0) := lim ρ→0 + ♭ m (G, ρ). We also intend β md (G, 0) := lim ρ→0 + β md (G, ρ), b md (G, 0) := lim ρ→0 + b md (G, ρ), if these quantities are finite for ρ close to zero.

2.6
Definition. Let n ∈ N. A map G as above is said to have the Π n property if: The main result on complex Nemytskij operators is the following.

2.7
Proposition. Let n, a ∈ N and a > d/2. Consider functions G : and Eq.s (2.31-2.34) hold again, with the definitions given therein for γ nd (G, ρ) and c nd (G, ρ). ⋄ The case of an x-independent G. Let us consider a function (derivable in the ordinary sense, as many times as required in the sequel); we regard G as an x-independent function of (u, x) or (z, x). Some of the previous formulae become simpler due to the vanishing of all derivatives ∇ m G (m ≥ 1) with respect to the x variables. More precisely, G has the Π n property for n ≥ 1 if and only if it is C n ; the Π 0 property is possessed by G if and only if G is C 0 and ♭ 0 (G, ρ) is finite and continuous in ρ.
Both in the real and in the complex case, we can study the Nemytskij operator

for any n if and only if
If this happens and G has the Π n property, Eq.(2.31) holds, taking in this case the form We shall exemplify this scheme in two cases: in the first G is a monomial in z and z, in the second it is the hyperbolic sine.
Examples. i) Let us put from here, one also infers that the only nonzero functions in (2.40) are the σ j0 for 1 ≤ j ≤ H + K, given by Thus, we have On the other hand, each coefficient ν j0 in the polynomial P m multiplies ρ j (see Eq. (2.18)). From here, one easily infers where the B J md are positive coefficients defined for any integer J ≥ 1, setting The final result on the Nemytskij operator A slightly weaker, but simpler bound can be obtained recalling that all the U coefficients are ≤ 1. For each m ≥ 1, this implies the equality at the beginning of the last line can be obtained directly from the relation (2.17) defining P m , to be applied at ξ = 0 to the function F (u, ξ) := u J . From (2.54) and (2.56) one readily infers Γ J nd ≤ J n for all n ≥ 0, whence Let us recall that, in the previous work [18], we have discussed some inequalities for the norm f g n of a product, implying directly an estimate of the form . In all the tests we have performed, the upper bounds of [18] on the constants K nad and the bounds Γ H+K nd S ad arising from (2.55), with H + K = 2, are very close numerically. A result of [18] also implies the weaker bound (2.57) for H + K = 2. For H + K > 2, an estimation of f H f K n could be performed by iteration of the bounds [18] on the product of two functions; however, the estimates derived in this elementary way would be sensibly rougher than (2.55) or (2.57).
The purpose of this Section and of the next two is to give some tools for proving the main results of the paper. Here we concentrate on some algebraic aspects of tensor calculus, on the differentiation of tensors and on a general formula for the derivatives of any order of a composite function G(f, x). In Sect.4 we will connect these results with the polynomials P m ; in Sect.5 we will be concerned with functional spaces of tensor-valued mappings, and some related inequalities. We already introduced the tensor powers ⊗ ℓ C d . For ℓ, m ∈ N, the tensor product operation for λ 1 , ..., µ m ∈ {1, ..., d} (for ℓ = 0, T ⊗ U means the product T U between the complex number T and U, and the same is understood for m = 0). The product ⊗ is clearly associative. Let ℓ ∈ N 0 and σ a permutation of {1, ..., ℓ}; then, we a have a (linear) permutation operator and introduce a symmetrisation operator where ℓ! stands for the set of all permutations of {1, ..., ℓ} (whose cardinality is ℓ!). Note that S is the identity map for ℓ = 1; we intend S to be the identity for ℓ = 0 also. A tensor T is symmetric if and only if ST = T . For ℓ, m ∈ N, the symmetrised tensor product operation ∨ is defined setting This operation is commutative and associative (see the Appendix) : for arbitrary tensors T, U as above and V ∈ ⊗ p C d . (Often, ∨ is considered to act on symmetric tensors only; this leads to the so-called symmetric tensor algebra, see, e.g., [20] [11]). If T ∈ ⊗ ℓ C d and q ∈ N, we will write intending this to be 1 if q = 0. We already defined the norm | | of tensors, see Eq.(2.2); for all T, U it is (see the Appendix). Sometimes, we need to employ the complex conjugate T of a tensor T ∈ ⊗ ℓ C d ; this has components We now pass to spaces of tensor-valued functions, or tensor fields the products ⊗, ∨, the conjugate , etc. are defined pointwisely on such functions.
is the space of mappings as in (3.11), where all components are C n in the usual sense. Given a tensor field T as above, of class C 1 , we define the derivative ∇T to be the tensor field of order ℓ + 1 with components For p ∈ N we denote with ∇ p the p-th power of the operator ∇ (acting on C p tensor fields); for any (of course derivatives ∂ λ ℓ+p , ... ∂ λ ℓ+1 commute, so their ordering is immaterial). In the sequel, we often employ the symmetrised derivative and denote with ∇ p S the p-th power of the operator ∇ S . On any function f ∈ C p (R d , C), we have ∇ p S f = ∇ p f = the already defined tensor field (2.6) . One can write a Leinitz rule for ∇ and the product ⊗, see the Appendix; we will not use this rule directly and always refer to a symmetrized version, also proved in the Appendix. The "symmetrized" Leibnitz rule reads What has been said up to now for complex tensors can be rephrased in the real case; so, we can define the product ⊗ between tensors T ∈ ⊗ ℓ R d , U ∈ ⊗ m R d , the symmetrisation operator S, the ∨ product and the derivatives ∇, ∇ S of real tensor fields, which satisfy again the previously mentioned Leibnitz rules.
Parameter dependent tensor fields. We begin from the real case. We consider tensor fields depending on a supplementary real variable u, i.e., mappings we define all previous operations ⊗, S, ∨, ∇, ∇ S treating u as a parameter, e.g., As in Sect.2, we denote with ∂ the derivative ∂/∂u. For any u-dependent tensor field T of order ℓ as in (3.17) and class C 1 , ∂T is the u-dependent tensor field, again of order ℓ, with components Of course ∂ commutes with ∇ and ∇ S . We can iterate the operations ∂, ∇, ∇ S ; so, for T of order ℓ and class C j+m , the tensor fields ∂ j ∇ m T , ∂ j ∇ m S T (both of order ℓ + m) are defined. The case of complex tensor fields, depending a complex parameter z, is worked out similarly. In this case we consider mappings any such map is said to be C n if its derivatives w.r.t. z, z and the real variables x, up to total order n, exist everywhere and are continuous. The z-dependent tensor fields are acted upon by the derivatives ∂ := ∂/∂z, ∂ := ∂/∂z, ∇, ∇ S and their iterates.
Iterated derivatives of composite functions. Again, we start from the real case. We consider a u-dependent tensor field of arbitrary order ℓ We associate to H a "tensor" Nemytskij operator sending a function f : R d → (−r, r) into the tensor field  H(f, x). Assume H, f to be C 1 ; then (recall ∂ := ∂/∂u). In compact form, this amounts to write we can symmetrise this result by application of S to both sides, the result being To generalize this to any order, we need some notations. Let us put such that for each dimension d, each C m function G : These coefficients are given by Remarks. i) For d = 1 and an x-independent G, this statement is known as the Faà di Bruno formula, see e.g. [5]; we acknowledge S. Paveri Fontana for bibliographical indications concerning this case. Of course, for d = 1 the ∨ product collapses into the usual product of real numbers. Our formulation in terms of tensor products holds for an arbitrary space dimension d (and x-dependent G). Some multidimensional extensions of the Faà di Bruno formula, not using the tensor language, were proposed in the literature (see, e.g., [8] [15]); another one was suggested to us by G. Meloni [14], to whom we are grateful for general discussions on this subject and its relations with the Stirling numbers. ii) Eq.(3.30) can be written in terms of the Bell polynomials and some differential operators. The Bell polynomials Y w (t 1 , t 2 , ..., t w ) [5] (which are very similar to the Schur polynomials [10]) are defined by the formal expansion exp ( ∞ r=1 t r α r /r!) = 1 + ∞ w=1 Y w (t 1 , t 2 , ..., t w ) α w /w!, and given explicitly by for each integer w ≥ 1. Comparing with Eq.s(3.30) (3.31), one finds that the formula for the derivatives of a composite function can be written in the symbolic form Let us pass to prove uniqueness of the coefficients, and derive the explicit expression (3.31); this will also make clear that P m|p = 0 for all p. Let us consider any family (P m|p ) fulfilling (3.30) for a given m and all d, G, f . We apply Eq.(3.30) with this set of coefficients and d = 1, calling ξ the x variable; the operator ∇ is just the ordinary derivative d/dξ, and (3.30) becomes (adding primes to indices j, ℓ, for future convenience) for all C m functions G : (−r, r) × R → R and f : R → (−r, r). Now, we choose a sequence p = (p 1 , p 2 , ...) in the domain of the family (3.29), and set j : where a s is an arbitrary real parameter if p s = 0, and a s := 0 if p s = 0. Let us evaluate at ξ = 0 all derivatives in (3.34). The only non vanishing derivatives (∂ j ′ +ℓ ′ G/∂u j ′ ∂ξ ℓ ′ ) at (f (0), 0) = (0, 0) occur for j ′ = j, ℓ ′ = ℓ and equal 1; furthermore (d s f /dξ s )(0) = a s . Thus, inserting these choices of G, f into Eq.(3.34) and setting ξ = 0, we obtain intending 0 0 := 1; this implies Computing the indicated coefficient in the above polynomial, one obtains for P m|p the expression (3.31); the calculation is based on the identity to be applied with z s := a s ξ s /s! .

⋄
We pass to the complex case, and consider a z-dependent tensor field then a tensor Nemytskij operator is defined, sending a function f : R d → B C (0, r) into the tensor field with ∇f denoting the complex conjugate of ∇f (recall Eq.(3.10)). The symmetrised version of this identity is for all integers m, h, k, w such that m ≥ 1, h, k ≥ 0, 1 ≤ h + k ≤ w. All sets D hkw are nonempty ( 4 ), and D m = ∪ 1≤w≤m, h,k≥0 3.2 Proposition. Let m ∈ N 0 . Then, there exists a unique family of positive, integer coefficients P m|pq (p,q)∈Dm (3.45) such that for each dimension d, each C m function G : These coefficients are given by Proof. It is similar to the proof of Prop.3.1. The existence of a set of nonnegative, integer coefficients fulfilling (3.46) for all d, G and f is proved by recursion over m. Now, let us consider for given m a family of coefficients (P m|pq ) fulfilling (3.46), and prove these coefficients to be uniquely determined as in Eq.(3.47), which also implies their positiveness. To this purpose, it suffices to consider Eq.(3.46) for d = 1, calling ξ the x variable (and adding primes to indices), which gives for all C m functions G : B C (0, r) × R → C and f : R → B C (0, r).  We begin with the Proof of Lemma 2.1 . Let us first prove the uniqueness of the polynomial fulfilling (2.17) for all functions F . To this purpose we consider any polynomial P fulfilling (2.17), and show that P ((ν hℓ ), ρ) is uniquely determined for all ν hℓ ∈ R and ρ ∈ (0, +∞), a fact yielding the thesis. To prove this fact, given (ν hℓ ), ρ as above we define a function F = F m,(ν hℓ ),ρ : Then, it is found that these results, with Eq.(2.17) for P , imply the r.h.s. is given by Eq.(4.3) for P = P m , considering the derivative at ξ = 0 and the function F m, and this proves Eq. (2.19). Finally, let us write (e ξ − 1) j = j r=1 j r e rξ (−1) j−r and expand e rξ about ξ = 0; inserting the expansion in the previous equation, and isolating the coefficient of ξ m we obtain the expression (2.20) for P mjℓ . ⋄ Now, let us point out a relation between the coefficients in the general formulae for the derivation of a composite function and the coefficients P mjℓ of the polynomials P m (an equivalent formulation of the forthcoming Eq.(4.5) for ℓ = 0 can be found in [5]).
We always consider spaces of complex tensor-valued functions; the real analogues can be regarded as subspaces of the complex ones.
Fourier transform and Sobolev spaces. We denote with F : the Fourier transform for tempered distributions; when using this, we write k = (k 1 , ..., k d ) for the running variable on R d and k : R d → R d for the identity mapping of this space into itself. We normalise F so that C). We extend componentwisely the Fourier transform to tensor valued tempered distributions, setting for each T as before. For f ∈ S ′ (R d , C) and m ∈ N, the Fourier representation of distributional derivatives can be written as where ⊗ m (ik) := (ik) ⊗ ... ⊗ (ik) (m times). This fact, with the preservation of the whenever one of the above three functions is in L 2 ; the second equality follows from the fact that the tensor norm | ⊗ m (ik)| equals |k| m = k 2 1 + ... + k 2 d m for each k ∈ R d (recall Eq.(3.9)). Due to these facts and to the equality we see from Eq.s(2.7) (2.8) that thus, the conventional definition of Bessel potential spaces is recovered (Eq.s(5.6) (5.7) also make sense for non integer n, and could be written in terms of the operator Hausdorff-Young inequality. We recall the definition (2.13) for the function E( ). Let p ∈ [1, 2], r ∈ [2, +∞] be such that 1/p + 1/r = 1. Further, let m ∈ N; for all (5.8) furthermore, the above constant is the sharp one for the written inequality. In the case m = 0, where T is C-valued, this result is classic (see Beckner's paper [3], or [12]; these give a slightly different expression for C rd due to a different normalisation for the Fourier transform). For m = 1, 2, 3, ..., the above statement follows from the Marcinkiewicz-Zygmund theorem on the componentwise vector extensions of operators between L p spaces (see, e.g., [7], Ch.5, Theor.2.7; it must be noted that the norm (2.2) we use on ⊗ m C d is a Hilbertian, ℓ 2 -type norm, as required by the theorem. We acknowledge L. Colzani for pointing out to us the cited result, and W. Beckner for general indications on the vectorial extensions of the Hausdorff-Young inequality).
An interpolation inequality. We rephrase in tensor notations a well known inequality; the aim is to point out the absence of d-dependent constants. and (intending ℓ/m := 1 for ℓ = m = 0 and 0 t := 0 for each t).
The second inequality (5.9) follows trivially from the first one, due to the Young A Gagliardo type inequality. The structure of the inequality is familiar, but little seems available about the constant appearing therein. Here we propose an explicit (probably non sharp) estimate for the constant, based on the use of Fourier methods and the Hausdorff-Young inequality (5.8). All derivatives in the forthcoming statement are intended in the distributional sense. and where E( ) is defined as in (2.13) and S ad is the constant (2.9).
Proof. We put provided that the last function be L p . To prove this and the rest, we write and apply the Hölder inequality to the above three factors, recalling that 1/s+1/s+ 1/r = 1/p; this gives Let us show that the norms of the three factors are finite, and compute them. We (some intermediate steps where integrals appear do not make sense for s or r = ∞, but the final results also cover these cases). Now, we insert Eq.s (5.17-5.19) into (5.16), and the result into Eq.(5.14); after explicitating r and s, we readily get the thesis (5.11). ⋄ The Adams-Frazier inequality. Again, we discuss an inequality whose structure is known [1] but for which the constants were not previously estimated, to the best of our knowledge. As in the previous subsection, we intend derivatives in the distributional sense.
Proof. The statement is trivial if h + k = 1; in the sequel h + k ≥ 2. We put for u ∈ {1, ..., h}, v ∈ {1, ..., k}; then p u , q v ∈ [2, +∞) and Thus, from the Hölder inequality we get provided that all norms in the r.h.s. be finite. In fact it is so, and the Gagliardo inequality with ℓ = i u or ℓ = g v and m as above gives Inserting these estimates into (5.23) and recalling that i 1 + ... + g k = m, we readily get under the given assumptions on G and f . Formula (3.46) for the derivatives of a composite function is essential for this purpose; the result is the following.
6.1 Proposition. Let n, a ∈ N and a > d/2. Consider a function G : B C (0, r) × R d → C with the Π n property, such that G(0, x) ∈ H n (R d , C), and a function where ♭ 0 , ♭ m , β md , b md are defined as in Eq.s(2.27), (2.29) and (2.30). Proof.
Step 2 (for n ≥ 1). It is ∇ m (G(f, x)) ∈ L 2 (R d , C) and (6.2) holds for 1 ≤ m ≤ n, The supplementary assumption that f be C n allows us to use Eq.(3.46) for the derivatives of a composite function, in the usual (i.e., not distributional) sense (next, the C n assumption on f will be removed by appropriate density arguments, see Step 3). For the sake of brevity, let us put then G also possesses the Π n property. By our assumptions, ∇ m (G(0, x)) is L 2 ; we will show that ∇ m (G(f, x)) is also L 2 , with its L 2 norm bounded by the r.h.s. of (6.2). Let us express ∇ m (G(f, x)) according to Eq.(3.46) (with G replaced by G); this implies provided that all norms in the r.h.s. be finite (we have used the Hölder inequality for tensor fields, in the form T ∨ S L 2 ≤ T L ∞ S L 2 ). Let h, k, ℓ be as in the above sum; then . Recalling the definition (2.39) we see that, for ρ ∈ [0, r), z ∈ B C (0, ρ) and x ∈ R d , due to (2.10), we can apply this inequality with z = f (x) and ρ = S ad f a , so Let us pass to the products with (p, q) ∈ D hk,m−ℓ ; any one of them can be reexpressed as . ≤ g k are positive integers, determined in an obvious way by the sequences (p, q). The above expression envolves h + k factors, with total order i 1 + ... + g k = m−ℓ, and its L 2 norm can be bounded by the Adams-Frazier inequality (5.20) in terms of ∇ m−ℓ f L 2 ; the last norm can be interpolated with the second inequality (5.9). Summing up Finally, let us consider the term ∇ m G(f, x) L 2 ; a straightforward generalization of Eq. (6.3) implies Let us insert the results (6.6) (6.7) (6.8) into Eq.(6.5); the sums over (q, p) ∈ D hk,m−ℓ of the coefficients P m|qp are related as in Eq.(4.6) to the coefficients P m,h+k,ℓ of the polynomial P m , and so Let us compare this result with the definition (2.40) of σ jℓ (G, ρ) in terms of the coefficients σ hkℓ (G, ρ), with h + k = j; this gives From the expression (2.18) of the polynomial P m and the definitions (2.29) (2.30) of β md , b md in terms of P m , we see that the previous equation means ∇ m (G(f, x)) L 2 ≤ r.h.s. of Eq.(6.2) , (6.11) as desired.
Step 3 (for n ≥ 1). It is ∇ m (G(f, x)) ∈ L 2 (R d , C) and (6.2) holds for 1 ≤ m ≤ n and f ∈ H n (R d , C) ∩ H a r (R d , R). Let f ∈ H n ∩ H a r , and put This is an L 2 function due to Step 1; the thesis follows if we show that ∇ m g is L 2 for 1 ≤ m ≤ n, with the L 2 norm bounded by the r.h.s. of (6.2). As anticipated, a density argument will be employed to prove this.
In the sequel, s is an index ranging over N, and s → indicates convergence for s → +∞. By standard density results, there is a sequence of functions (f s ) such that Let us put g s := G(f s , x) − G(0, x) ; (6.14) then, due to Steps 1 and 2, we have the statements on the limits over s follow from the continuity of the mappings ♭ 0 , ♭ m , b md , β md , and also imply sup s g s L 2 < +∞ , sup s ∇ m g s L 2 < +∞ (1 ≤ m ≤ n) . (6.18) Let us return to the function g of Eq.(6.12), which is also L 2 due to Step 1; we claim that x); to estimate this difference, we note that, by the Lagrange formula, for any radius ρ ∈ [0, r) and all z, whence |G(z ′ , x) − G(z, x)| ≤ (σ 100 (G, ρ) + σ 010 (G, ρ)) |z ′ − z| . (6.20) In particular, we choose ρ such that S ad f a < ρ and S ad f s a < ρ for s large; applying Eq.(6.20) with z = f (x) and z ′ = f s (x), and integrating (the squares of) both sides over x, we get thus proving (6.19). The rest of our argument depends on some known facts on weak convergence ( 5 ). Of course, Eq.(6.19) implies According to a general result on Sobolev spaces (see e.g. [6], Part 1, Sect.6, Lemma 6.2), statements (6.18) (6.21) are sufficient to infer that for 1 ≤ m ≤ n 5 Given a Banach space X, a sequence (u s ) with elements in X and another element u of X, it is said that u s s → u weakly in X if < α, u s > s →< α, u > for each α in the dual Banach space X ′ . By the uniform boundedness theorem, this implies sup s u s X < +∞ and u X ≤ lim inf s u s X , see e.g. [21]. The use of weak convergence in relation to Nemytskij operators is suggested in [19]. (6.29) inserting (6.28) (6.29) into (6.27), we obtain the bound (2.31-2.33) on G(f, x) − G(0, x) n ; the weaker bound (2.34) follows trivially, since f L 2 ≤ f n . ⋄ Comments and acknowledgments. The quantitative estimates in the present work are extended with little effort to Nemytskij operators on spaces H n (R d , R δ or C δ ), where δ is an arbitrary integer; here, this has not been done to keep notations as simple as possible. We again acknowledge W. Beckner, L. Colzani, G. Meloni and S. Paveri Fontana for useful indications. This work was partly supported by the GNFM of Istituto Nazionale di Alta Matematica and by MURST.
A Appendix. Some facts about tensors.

5.7).
Proofs of the commutative and associative properties (3.7) of the symmetrized tensor product. The arguments are standard [11], and reported only for completeness. Let T ∈ ⊗ ℓ C d , U ∈ ⊗ m C d . It is evident that U ⊗ T = P ζ (T ⊗ U), where ζ ∈ (ℓ + m)! is the permutation such that ζ (1)  Proof of Eq.(3.9) for the tensor norms. Consider tensors T ∈ ⊗ ℓ C d , U ∈ ⊗ m C d . The equality |T ⊗ U| 2 = |T | 2 |U| 2 is checked in an elementary way, explicitating the definitions of | | and ⊗. Furthermore, the last passage following from the definition of S and the triangular inequality for | |. On the other hand, it is clear from the very definition that each operator P σ is isometric w.r.t. the norm | |; so |P σ (T ⊗U)| = |T ⊗U| = |T ||U| for all σ ∈ (ℓ + m)!, and this implies |T ∨ U| ≤ |T ||U|, as desired.