In this paper we establish the ${{L}^{p}}$ boundedness of a class of singular integrals with rough kernels associated to polynomial mappings.

In this paper the authors establish the $L^p$ boundedness for several classes of Marcinkiewicz integral operators with kernels satisfying a condition introduced by Grafakos and Stefanov in Indiana Univ. Math. J.47 (1998), 455–469.

AMS 2000 Mathematics subject classification: Primary 42B25; 42B99

A systematic treatment is given of several classes of singular integrals. Their $L^p$ boundedness is proved when their kernels are given by functions $\Omega$ in $L\log L({\bf S}^{n-1})$ .

For n[ges ]2, let S(Rn) be the space of Schwartz class functions. The dual space of S(Rn) is denoted by S′(Rn) and is called the tempered distributions. For f∈S(Rn), define fˆ(w)=∫Rne−iw·xf(x)dx and for T∈S′(Rn), define Tˆ∈S′(Rn) by the formula 〈Tˆ, f〉=〈T, fˆ〉 for all f∈S(Rn). For 1[les ]p[les ]∞, let FLp(Rn)= {T∈S′(Rn)[ratio ] Tˆ∈Lp(Rn)}. Let β be a multi-index of nonnegative integers, β=(β1, β2, …, βn−1) with [mid ]β[mid ]= β1+β2+…+βn−1. For a nonnegative integer k and bounded Borel measures σβ on a compact subset K of Rn, [mid ]β[mid ][les ]k, we define T∈S′(Rn) by the formula

formula here

If T is given by (1), then we know that supp(T)⊂K and the order of T is at most k. In particular, when k=0, we say that T is a bounded measure on K. Denote EK(Rn)= {T∈S′(Rn)[ratio ]T is a bounded measure on K} and DK(Rn)= {T∈S′(Rn)[ratio ] supp(T)∪K}. If in addition we assume some connectness property of K, then one can show ([5, th. 2·3·10]) that if T∈DK(Rn), then T does have the representation (1) for some k[ges ]0. The question is when we can say that k=0. From the Hausdorff–Young inequality and the proof of theorem 7·6·6 in [5], one sees that if the interior of K (in Rn) is not empty, then DK(Rn) ∩FLp(Rn) ⊂EK(Rn) if and only if 1[les ]p[les ]2. Much remains to be understood if the interior of K is empty.

In this paper, we seek to prove a similar result for compact subsets of hypersurfaces of Rn, motivated from the differential equations with constant coefficients.

We use a simple approximation method to prove the Holder continuity of the generalized de Rham functions.

1. Consider the following dilatation equation

where |α|<l/2. Suppose that f is an integrable solution of (1); then f must satisfy

where is the Fourier transform of f, and

which immediately leads to

In this paper we study the uniform boundedness of oscillatory singular integral operators with degenerate phase functions on the Hardy space H1. The H1 boundedness was previously known when the phase function is nondegenerate. Here we obtain a sufficient condition for H1 boundedness which allows the phase function vanishing to infinite order.

Let k ∈ R, k ≥ 3. Consider the curve of class C3 in R3 defined by

Let σk(t) = t(k−3)/6dt be the affine arclength measure on γk. Define the convolution operator T (= Tk) by

Let λ be a certain measure supported on a curve in R3. We prove that if the curve has nonzero curvature and torsion, then λ * L3/2 ⊂ L2.

Consider the operator

where , and K is smooth away from the origin and homogeneous of degree —μ, for some μ ≥ 0. The following theorem was proved in [4].