A variant of Hardy's and Miyachi's theorems for the Bessel–Struve transform

In this paper, we establish new versions of Hardy's and Miyachi's theorems for the Bessel–Struve transform.     

Unique continuation for solutions to pde's; between hörmander's theorem and holmgren' theorem

We prove a new unique continuation result for solutions to partial differential equations, “interpolating” between Holmgren's Theorem and Hörmander's Theorem. More precisely, under some partial analyticity assumptions on the coefficients we obtain an intermediate unique continuation result which is in general weaker than Holmgren's Theorem (which applies to problems with analytic coefficients) but stronger than Hörmander's Theorem (which applies to problems with C¹ coefficients). Some applications to the wave and the Schroedinger equation are considered next. In particular we obtain a result conjectured by Hörmander, namely that for the wave equation with C¹ but time independent coefficients one has unique continutaion across any noncharacteristic surface.     

Abstract Cauchy problem for the Bessel–Struve equation

We consider the Cauchy problem for the Bessel–Struve equation in a Banach space. A sufficient condition for the solvability of this problem is proved, and the solution operator is written in explicit form via the Bessel and Struve operator functions. A number of properties is established for the solutions.     

An analogue of Beurling’s theorem for the Jacobi transform

In this paper, we prove Beurling＇s theorem for the Jacobi transform, from which we derive some other versions of uncertainty principles.     

The A-integral and restricted Ahlfors–Beurling transform

In this paper we prove that the restricted Ahlfors–Beurling transform of a Lebesgue integrable function is A-integrable and derive an analogue of Riesz's equality holds.     

A Generalization of Hörmander's Inequality-I

We give sufficient conditions to generalize Hörmander's inequality to the case of operators with multiple characteristics of order higher than two     

On the Generalization of Hörmander's Inequality

ABSTRACT

We are concerned with a lower bound with a gain of k/2 + 1 derivatives for the class OP N ^{m, k} (X, Σ) of pesudodifferential operators with characteristics of even multiplicity k ≥ 2. In the case of double characteristics operators (k = 2), we recapture a well-known inequality due to Hörmander.     

The Ahlfors–Beurling transform on Morrey spaces with variable exponents

We establish the vector-valued inequalities of the Ahlfors–Beurling operator on Morrey spaces with variable exponents. As consequences of these inequalities, we have the boundedness of the Ahlfors–Beurling transform on Lebesgue spaces with variable exponents and Morrey spaces. The results obtained in this paper are new in the case of Morrey spaces.     

A Wiener–Wintner theorem for the Hilbert transform

We prove the following extension of the Wiener–Wintner theorem and the Carleson theorem on pointwise convergence of Fourier series: For all measure-preserving flows (X,μ,T _t ) and f∈L ^p (X,μ), there is a set X _f ⊂X of probability one, so that for all x∈X _f ,

A Nash Type Result for Divergence Parabolic Equation Related to Hörmander's Vector Fields

In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result, i.e., the local Hölder regularity for weak solutions. After deriving the parabolic Sobolev inequality, (1,1) type Poincaré inequality of Hörmander's vector fields and a De Giorgi type Lemma, the Hölder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the isomorphism between parabolic Campanato space and parabolic Hölder space. As a consequence, we give the Harnack inequality of weak solutions by showing an extension property of positivity for functions in the De Giorgi class.     

Paley–Wiener–Schwartz type theorem for the wavelet transform

In the present paper, we discuss about extension of the wavelet transform on distribution space of compact support and develop the Paley–Wiener–Schwartz type theorem for the wavelet transform on the same. Furthermore, Paley–Wiener–Schwartz type theorem for the wavelet transform is also established using the relation between the wavelet transform and double Fourier transform.     

Explicit estimates of sums related to the Nyman–Beurling criterion for the Riemann Hypothesis

We give an estimate for sums appearing in the Nyman–Beurling criterion for the Riemann Hypothesis. These sums contain the Möbius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably sharp in comparison to other sums containing the Möbius function. The bound is smaller than the trivial bound – essentially the number of terms – by a fixed power of that number. The exponent is made explicit. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.     

Converse Sturm–Hurwitz–Kellogg theorem and related results

We prove that if Vⁿ is a Chebyshev system on the circle and f is a continuous real-valued function with at least n + 1 sign changes then there exists an orientation preserving diffeomorphism of S¹ that takes f to a function L²-orthogonal to V. We also prove that if f is a function on the real projective line with at least four sign changes then there exists an orientation preserving diffeomorphism of that takes f to the Schwarzian derivative of a function on . We show that the space of piecewise constant functions on an interval with values ± 1 and at most n + 1 intervals of constant sign is homeomorphic to n-dimensional sphere. To V. I. Arnold for his 70th birthday     

Paley–Wiener theorem for the Weinstein transform and applications

In this paper our aim is to establish the Paley–Wiener Theorem for the Weinstein Transform. Furthermore, some applications are presents. In particular some properties for the generalized translation operator associated with the Weinstein operator are proved, an integral representation and a series representation for a function in the Paley–Wiener classes are investigated.     

The trace problem for vector fields satisfying Hörmander's condition

Trace theorems are proved for non-isotropic Sobolev and -Lipschitz spaces defined by vector fields satisfying H?rmander's bracket condition of order 2. It is shown that the loss of regularity by traces is the same as in the classical case. Received March 17, 1997; in final form April 18, 1998     

A Jordan–Hölder theorem for weakly group-theoretical fusion categories

We prove a version of the Jordan–Hölder theorem in the context of weakly group-theoretical fusion categories. This allows us to introduce the composition factors and the length of such a fusion category \({\mathcal C}\), which are in fact Morita invariants of \({\mathcal C}\).