The Hardy–Littlewood–Sobolev inequality for (β,γ)-distance Riesz potentials (original) (raw)

The Stein-Weiss type inequalities for the B-Riesz potentials

Journal of Mathematical Inequalities, 2011

We establish two inequalities of Stein-Weiss type for the Riesz potential operator I α,γ (B− Riesz potential operator) generated by the Laplace-Bessel differential operator Δ B in the weighted Lebesgue spaces L p,|x| β ,γ. We obtain necessary and sufficient conditions on the parameters for the boundedness of I α,γ from the spaces L p,|x| β ,γ to L q,|x| −λ ,γ , and from the spaces L 1,|x| β ,γ to the weak spaces W L q,|x| −λ ,γ. In the limiting case p = Q/α we prove that the modified B− Riesz potential operator I α,γ is bounded from the spaces L p,|x| β ,γ to the weighted B − BMO spaces BMO |x| −λ ,γ. As applications, we get the boundedness of I α,γ from the weighted B-Besov spaces B s pθ ,|x| β ,γ to the spaces B s qθ ,|x| −λ ,γ. Furthermore, we prove two Sobolev embedding theorems on weighted Lebesgue L p,|x| β ,γ and weighted B-Besov spaces B s pθ ,|x| β ,γ by using the fundamental solution of the B-elliptic equation Δ α/2 B .

The Hardy–Littlewood–Sobolev theorem for Riesz potential generated by Gegenbauer operator

Transactions of A. Razmadze Mathematical Institute, 2016

In this paper we introduced and studied the maximal function (G-maximal function) and the Riesz potential (G-Riesz potential) generated by Gegenbauer differential operator G λ =  x 2 − 1  1 2 −λ d d x  x 2 − 1  λ+ 1 2 d d x. The L p,λ boundedness of the G-maximal operator is obtained. Hardy-Littlewood-Sobolev theorem of G-Riesz potential on L p,λ spaces is established. c

Inequalities associated to Riesz potentials and non-doubling measures with applications

2010

The main focus of this work is to study the classical Calderón-Zygmund theory and its recent developments. An attempt has been made to study some of its theory in more generality in the context of a nonhomogeneous space equipped with a measure which is not necessarily doubling. We establish a Hedberg type inequality associated to a non-doubling measure which connects two famous theorems of Harmonic Analysis-the Hardy-Littlewood-Weiner maximal theorem and the Hardy-Sobolev integral theorem. Hedberg inequalities give pointwise estimates of the Riesz potentials in terms of an appropriate maximal function. We also establish a good lambda inequality relating the distribution function of the Riesz potential and the fractional maximal function in (R, dμ), where μ is a positive Radon measure which is not necessarily doubling. Finally, we also derive potential inequalities as an application. INEQUALITIES ASSOCIATED TO RIESZ POTENTIALS AND NON-DOUBLING MEASURES WITH APPLICATIONS by MUKTA BAHA...

Some Embe dding Theorems and Properties of Riesz Potentials

American Journal of Mathematics and Statistics, 2013

It is well known that potential type integrals play an important role in research of the partial differential equations. The present paper studies properties of Riesz potential in terms of local oscillation of functions.

On the Hardy–Sobolev–Maz'ya Inequality and Its Generalizations

International Mathematical Series, 2009

The paper deals with natural generalizations of the Hardy-Sobolev-Maz'ya inequality and some related questions, such as the optimality and stability of such inequalities, the existence of minimizers of the associated variational problem, and the natural energy space associated with the given functional.

Isoperimetric inequalities for Schatten norms of Riesz potentials

Journal of Functional Analysis, 2016

In this note we prove that the ball is a maximiser of some Schatten p-norms of the Riesz potential operators among all domains of a given measure in R d. In particular, the result is valid for the polyharmonic Newton potential operator, which is related to a nonlocal boundary value problem for the poly-Laplacian extending the one considered by M. Kac in the case of the Laplacian, so we obtain and isoperimetric inequalities for its eigenvalues as well, namely, analogues of Rayleigh-Faber-Krahn and Hong-Krahn-Szegö inequalities.

CHARACTERIZATION OF RIESZ AND BESSEL POTENTIALS ON VARIABLE LEBESGUE SPACES

2006

Riesz and Bessel potential spaces are studied within the framework of the Lebesgue spaces with variable exponent. It is shown that the spaces of these potentials can be characterized in terms of convergence of hypersingular integrals, if one assumes that the exponent satisfies natural regularity conditions. As a consequence of this characterization, we describe a relation between the spaces of Riesz or Bessel potentials and the variable Sobolev spaces.