Journal of Fourier Analysis and Applications最新文献

Let (Omega ) be a homogeneous function of degree zero, have vanishing moment of order one on the unit sphere (mathbb {S}^{d-1})((dge 2)). In this paper, our object of investigation is the following rough non-standard singular integral operator

where A is a function defined on ({mathbb {R}}^d) with derivatives of order one in ({textrm{BMO}}({mathbb {R}}^d)). We show that (T_{Omega ,,A}) enjoys the endpoint (Llog L) type estimate and is (L^p) bounded if (Omega in L(log L)^{2}({mathbb {S}}^{d-1})). These results essentially improve the previous known results given by Hofmann (Stud Math 109:105–131, 1994) for the (L^p) boundedness of (T_{Omega ,,A}) under the condition (Omega in L^{q}({mathbb {S}}^{d-1})) ((q>1)), Hu and Yang (Bull Lond Math Soc 35:759–769, 2003) for the endpoint weak (Llog L) type estimates when (Omega in textrm{Lip}_{alpha }({mathbb {S}}^{d-1})) for some (alpha in (0,,1]).

In this paper we investigate the harmonic analysis of infinite convolutions generated by admissible pairs on Euclidean space ({{mathbb {R}}}^n). Our main results give several sufficient conditions so that the infinite convolution (mu ) to be a spectral measure, that is, its Hilbert space (L^2(mu )) admits a family of orthonormal basis of exponentials. As a concrete application, we give a complete characterization on the spectral property for certain infinite convolution on the plane ({{mathbb {R}}}^2) in terms of admissible pairs.

In this article, we studied the convolution operators (M_k) with oscillatory kernel, which are related to the solutions of the Cauchy problem for the strictly hyperbolic equations. The operator (M_k) is associated to the characteristic hypersurfaces(Sigma subset {mathbb {R}}^3) of a hyperbolic equation and smooth amplitude function, which is homogeneous of the order (-k) for large values of the argument. We investigated the convolution operators assuming that the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point (vin Sigma ) at which, exactly one of the principal curvatures of the surface (Sigma ) does not vanish. Such surfaces exhibit singularities of the type A in the sense of Arnold’s classification. Denoting by (k_p) the minimal number such that (M_k) is (L^pmapsto L^{p'})-bounded for (k>k_p,) we showed that the number (k_p) depends on some discrete characteristics of the surface (Sigma ).

We prove an extrapolation of compactness theorem for operators on Banach function spaces satisfying certain convexity and concavity conditions. In particular, we show that the boundedness of an operator T in the weighted Lebesgue scale and the compactness of T in the unweighted Lebesgue scale yields compactness of T on a very general class of Banach function spaces. As our main new tool, we prove various characterizations of the boundedness of the Hardy-Littlewood maximal operator on such spaces and their associate spaces, using a novel sparse self-improvement technique. We apply our main results to prove compactness of the commutators of singular integral operators and pointwise multiplication by functions of vanishing mean oscillation on, for example, weighted variable Lebesgue spaces.

We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by finite homogeneous Sobolev norm together with finite (L^2) norm of the Riesz potentials. As a byproduct we prove also existence of maximizers for the interpolation inequalities in Sobolev spaces for radially symmetric fractional super and sub harmonic functions.

The main result of this note is the strengthening of a quite arbitrary a priori Fourier restriction estimate to a multi-parameter maximal estimate of the same type. This allows us to discuss a certain multi-parameter Lebesgue point property of Fourier transforms, which replaces Euclidean balls by standard ellipsoids or axes-parallel rectangles. Along the lines of the same proof, we also establish a d-parameter Menshov–Paley–Zygmund-type theorem for the Fourier transform on ({mathbb {R}}^d). Such a result is interesting for (dgeqslant 2) because, in a sharp contrast with the one-dimensional case, the corresponding endpoint ({text {L}}^2) estimate (i.e., a Carleson-type theorem) is known to fail since the work of C. Fefferman in 1970. Finally, we show that a Strichartz estimate for a given homogeneous constant-coefficient linear dispersive PDE can sometimes be strengthened to a certain pseudo-differential version.

Revisiting the main point of the almost everywhere convergence, it becomes clear that a weak (1,1)-type inequality must be established for the maximal operator corresponding to the sequence of operators. The better route to take in obtaining almost everywhere convergence is by using the uniform boundedness of the sequence of operator, instead of using the mentioned maximal type of inequality. In this paper it is proved that a sequence of operators, defined by matrix transforms of the Walsh–Fourier series, is convergent almost everywhere to the function (fin L_{1}) if they are uniformly bounded from the dyadic Hardy space (H_{1} left( {mathbb {I}}right) ) to (L_{1}left( mathbb {I}right) ). As a further matter, the characterization of the points are put forth where the sequence of the operators of the matrix transform is convergent.

We consider the initial value problem (IVP) associated to the cubic nonlinear Schrödinger equation with third-order dispersion

for given data in the Sobolev space (H^s(mathbb R)). This IVP is known to be locally well-posed for given data with Sobolev regularity (s>-frac{1}{4}) and globally well-posed for (sge 0) (Carvajal in Electron J Differ Equ 2004:1–10, 2004). For given data in (H^s(mathbb R)), (0>s> -frac{1}{4}) no global well-posedness result is known. In this work, we derive an almost conserved quantity for such data and obtain a sharp global well-posedness result. Our result answers the question left open in (Carvajal in Electron J Differ Equ 2004:1–10, 2004).

Let G be a compact group and G/H a homogeneous space where H is a closed subgroup of G. Define an operator (T_H:C(G) rightarrow C(G/H)) by (T_Hf(tH)=int _H f(th) , dh) for each (tH in G/H). In this paper, we extend (T_H) to a norm-decreasing operator between (L^p)-spaces with a vector measure for each (1 le p <infty ). This extension will be used to derive properties of invariant vector measures on G/H. Moreover, a definition of the Fourier transform for (L^p)-functions with a vector measure is introduced on G/H. We also prove the uniqueness theorem and the Riemann–Lebesgue lemma.

Abstract

We prove an (L^p) -spectral multiplier theorem for sub-Laplacians on Heisenberg type groups under the sharp regularity condition (s>dleft| 1/p-1/2right| ) , where d is the topological dimension of the underlying group. Our approach relies on restriction type estimates where the multiplier is additionally truncated along the spectrum of the Laplacian on the center of the group.