Logarithmic Bounds for Oscillatory Singular Integrals on Hardy Spaces

It is well known that T P is bounded from L(R) to L(R) when 1 < p < ∞ and also from L(R) to L(R). Additionally, L → L and L → L bounds are dependent on the degree of the phase polynomial P only, not its coefficients (see [1, 2]). However, for H(R) → H(R) boundedness of T P , the answers are not nearly as clear-cut. First, it was shown in [3] that, in general, T P may fail to be bounded on H(R) and when the coefficients of the first-order terms of P vanish, T P is bounded from H(R) to itself with a bound independent of the higher order coefficients of P. More recent work can be found in [4, 5], including the following.


Introduction
For  ∈ N, let () be a Calderón-Zygmund kernel on R  and let () be a polynomial of  variables with real coefficients.Consider the following oscillatory singular integral operator: (−)  ( − )  () . ( It is well known that   is bounded from   (R  ) to   (R  ) when 1 <  < ∞ and also from  1 (R  ) to  1,∞ (R  ).Additionally,   →   and  1 →  1,∞ bounds are dependent on the degree of the phase polynomial  only, not its coefficients (see [1,2]).However, for  1 (R  ) →  1 (R  ) boundedness of   , the answers are not nearly as clear-cut.First, it was shown in [3] that, in general,   may fail to be bounded on  1 (R  ) and when the coefficients of the first-order terms of  vanish,   is bounded from  1 (R  ) to itself with a bound independent of the higher order coefficients of .
More recent work can be found in [4,5], including the following.
Theorem 1 (see [5]).Let  ∈ N,  ≥ 2, and () = ∑ 0≤||≤     be a polynomial of degree  in R  with real coefficients.Let  be a Calderón-Zygmund kernel and let   be given as in (1).Then, there exists a positive constant  such that for all  ∈  1 (R  ).The constant  may depend on , , and  but is independent of the coefficients {  } of .
In order to determine the optimal bound on ‖  ‖  1 → 1 , an example was given in [5] to show that, as hold for all  ∈  1 (R  )?
In this paper, we will prove that the answer to the above question is affirmative for all quadratic polynomials.Namely, we have the following.Theorem 2. Let  ∈ N and () = ∑ 0≤||≤2     be a quadratic polynomial in R  with real coefficients.Let  be a Calderón-Zygmund kernel and let   be given as in (1).Then, there exists a positive constant  such that for all  ∈  1 (R  ).The constant  may depend on  and  but is independent of the coefficients {  } of .
We point out that  denotes an absolute constant whose value may change from line to line.

Some Definitions and Lemmas
Many of the tools we use are known.For readers who wish to see the definitions and some of their properties, the following references are suggested: [6][7][8][9][10][11][12].
Let  be a function in the Schwartz space S(R  ) such that ∫ R  () = 1.For each  ∈  1 loc (R  ) and  ∈ R  , we let where   () =  − (/).
Lemma 8. Let  ∈ N and () = ∑ 0≤||≤2     be a quadratic polynomial in R  with real coefficients.Let  be a Calderón-Zygmund kernel satisfying ( 11)-( 12) and let   be given as in (1).Then, there exists a positive constant  such that for every  1 atom  which satisfies ( 7)-( 9) with  = 0 and  = 1.The constant  may depend on  and  but is independent of {  }, , and .
In this case, we have In this case, we let It follows from Theorem For  ∈ R  and  ∈ (0, 1), we have        (−) − Thus, (27) holds in both cases.

Proof of Main Theorem
To finish the proof, we recall the following result concerning Riesz transforms and Hardy spaces.
We will now give the proof of Theorem 2.