Optimal Regularity Properties of the Generalized Sobolev Spaces

where C, γ > 0, is the Hölder-Zygmund space (see [1]). In the critical case k = n/p the function f ∈ W p may not be even continuous. The result (1) is not optimal. We prove that the optimal one is obtained if in (1) L is replaced by the Marcinkiewicz space L. In this paper we prove similar optimal results, when L is replaced by a more general rearrangement invariant space E. The Sobolev space WE consists of all f ∈ W 1 with a finite quasinorm ‖f‖ WE =


Introduction
The classical Sobolev space    , 1 ≤  < ∞, consists of all locally integrable functions , defined on R  ,  ≥ 1, with the Lebesgue measure, such that the following norm is finite: ‖‖    = ∑ ||≤ ‖  ‖  , where ‖‖  stands for the   -norm.In investigating the regularity of the function  ∈    , we may assume, without any loss of generality, that  ∈  1 (Ω), Ω is a domain in R  , and  is zero outside Ω.For simplicity we suppose that the Lebesgue measure of Ω equals one and that the origin lies in Ω.It is well known that in the supercritical case  > /, where C  ,  > 0, is the Hölder-Zygmund space (see [1]).
In the critical case  = / the function  ∈    may not be even continuous.The result (1) is not optimal.We prove that the optimal one is obtained if in (1)   is replaced by the Marcinkiewicz space  ,∞ .In this paper we prove similar optimal results, when  ,∞ is replaced by a more general rearrangement invariant space .The Sobolev space    consists of all  ∈    1 with a finite quasinorm ‖‖    = ∑ ||≤ ‖  ‖  .More precisely, we consider quasinormed rearrangement invariant spaces , consisting of functions  ∈  1 (Ω), such that the quasinorm ‖‖  ≈   ( * ) < ∞, where   is a monotone quasinorm, defined on  + with values in [0, ∞] and  + is the cone of all locally integrable functions  ≥ 0 on (0, 1) with the Lebesgue measure.Monotonicity means that  1 ≤  2 implies   ( 1 ) ≤   ( 2 ).We suppose that  ∞ (Ω) →  →  1 (Ω), which means continuous embeddings.Here  * is the decreasing rearrangement of , given by  * () = inf{ > 0 :   () ≤ },  > 0, and   is the distribution function of , defined by   () = |{ ∈ R  : |()| > }|  , | ⋅ |  denoting Lebesgue -measure.Note that  * () = 0 for  > 1.Finally,  * * () := (1/) ∫  0  * ().Let   ,   be the Boyd indices of .For example, if  =   , then   =   = 1/ and the condition / ≥ 1/ means / ≥   > 0. Note that for  >  this is always satisfied.For these reasons we suppose that for the general , 0 <   =   ≤ 1 and the case min(, )/ >   is called super-critical, while the case min(, )/ =   -critical.In the super-critical case the function  ∈    is always continuous, while the spaces in the critical case   = /,  < , can be divided into two subclasses: in the first subclass the functions  ∈    may not be continuous-then the target space is rearrangement invariant, while these functions in the second subclass are continuous and the target space is the generalized Hölder-Zygmund space C (see Definition 1).The separating space for these two subclasses is given by the Lorentz space  /,1 ,  < .If  ≥ ; then    consists of continuous functions (see the classical result of Stein [2]).
The main goal of this paper is to prove optimal embeddings of the Sobolev space    into the generalized Hölder-Zygmund space C.First we prove that this embedding for  ≤  is equivalent to the continuity of the operator   () = ∫  0  /−1 ().The case  >  is reduced to the continuity of   by using the lifting principle ( [1]).Moreover, if, for example,  ≤ , then in the super-critical case, we can replace   by the operator of multiplication  / ().This implies a very simple characterization of both optimal target space  and optimal domain space .Namely, the quasinorm in the optimal target space () is given by   ( −/ ()) and the quasinorm in the optimal domain space () is given by   ( / ()).Note that we do not require   to be rearrangement invariant.In the critical case, the formula for the optimal target space is more complicated.In some cases it can be simplified.To this end, we apply the Σ method of extrapolation ( [3]) from the super-critical case.As a byproduct, we also characterize the embedding    →   ,  < , where   consists of all functions with bounded and uniformly continuous derivatives up to order .Namely, this is equivalent to the embedding  →  /(−),1 if  ≤ .The embedding  +  →   is always true since    →    1 →  0 .The problem of the optimal target rearrangement invariant space for potential type operators is considered in [4] by using   -capacities.The problem of the mapping properties of the Riesz potential in optimal couples of rearrangement invariant spaces is treated in [5][6][7].The optimal embeddings of generalized Sobolev type spaces into rearrangement invariant spaces are characterized in several papers [5,[8][9][10][11][12][13][14][15][16][17][18][19][20][21].The characterization of the continuous embedding of the generalized Bessel potential spaces into the generalized Hölder-Zygmund spaces C, when  is a weighted Lebesgue space, is given in [22].The optimal embeddings of Calderón spaces into the generalized Hölder-Zygmund spaces are characterized in [23].
The plan of the paper is as follows.In Section 2 we provide some basic definitions and known results.In Section 3 we characterize the embedding    → C.The optimal quasinorms are constructed in Section 4.

Preliminaries
We use the notations  1 ≲  2 or  2 ≳  1 for nonnegative functions or functionals to mean that the quotient  1 / 2 is bounded; also,  1 ≈  2 means that  1 ≲  2 and  1 ≳  2 .We say that  1 is equivalent to Let  be a quasinormed rearrangement invariant space as in the Introduction.There is an equivalent quasinorm   ≈   that satisfies the triangle inequality    ( 1 +  2 ) ≤    ( 1 ) +    ( 2 ) for some  ∈ (0, 1] that depends only on the space  (see [24]).We say that the quasinorm   satisfies Minkowski's inequality if for the equivalent quasinorm   , Usually we apply this inequality for functions   ∈  + with some kind of monotonicity.
In order to introduce the Hölder-Zygmund class of spaces, we denote the modulus of continuity of order  by where Δ  ℎ  are the usual iterated differences of .When  = 1 we simply write (, ).
Let  be a quasinormed space of locally integrable functions on the interval (0, 1) with the Lebesgue measure, continuously embedded in  ∞ (0, 1) and ‖‖  =   (||), where   is a monotone quasinorm on  + which satisfies Minkowski's inequality.The dilation function generated by   is given by where The choice of the space   is motivated by the fact that   ( Here  (,) , 0 <  <  < ∞, is the characteristic function of the interval (, ).
We will use the following equivalent quasinorm.
Definition 7 (optimal domain quasi-norm).Given the target quasinorm   ∈   , the optimal domain quasi-norm, denoted by  () , is the weakest domain quasi-norm; that is, for any domain quasinorm   ∈   such that the couple   ,   is admissible.
Definition 8 (optimal couple).The admissible couple   ∈   ,   ∈   is said to be optimal if both   and   are optimal.

Admissible Couples
Here we give a characterization of all admissible couples   ∈   ,   ∈   .We start with the main estimate.For  = 1, see also [26].
Example 20 and   ∈   and let  ≥ .Using Remark 16, we can construct an optimal domain  = Γ  (V) and this couple is optimal.Also   =   = 1 if V is slowly varying.
Theorem 22.The target quasinorm  () belongs to   , the couple   ,  () is admissible, and the target quasinorm is optimal.
In the critical case we do not know how to simplify the optimal target quasi-norm, defined in (74).Instead, we can construct a large class of domain quasinorms and the corresponding optimal target quasinorms by using extrapolation from the super-critical case.Recall some basic definitions and results from the extrapolation theory [3].Let ( 0 ,  1 ) be a couple of quasi-Banach spaces.The sigma extrapolation space Σ  (()( 0 ,  1 ) () − , ), -positive weight, 0 <  <  0 , 0 <  ≤ ∞, -positive decreasing weight, consists of all formula of Holmstedt for the -functional [25, page 310].Now we discuss the embedding    →  0 .For  = 1 more general results are proved in [27, Chapter 4].