Conjugate Exponents Research Articles

The Improvement of Hölder’s Inequality with r -Conjugate Exponents Relative to the L p -Norm

This paper investigates Hölder’s inequality under the condition of r -conjugate exponents in the sense that ∑ k = 1 s 1 / p k = 1 / r . Successively, we have, under r -conjugate exponents relative to the L p -norm, investigated generalized Hölder’s inequality, the interpolation of Hölder’s inequality, and generalized s -order Hölder’s inequality which is an expansion of the known Hölder’s inequality.

Radially Symmetric Solutions for Quasilinear Elliptic Equations Involving Nonhomogeneous Operators in an Orlicz-Sobolev Space Setting

We investigate the following elliptic equations: $$\left\{ {\matrix{ { - M\left( {\int_{{\mathbb{R}^N}} {\phi ({{\left| {\nabla u} \right|}^2}){\rm{d}}x} } \right){\rm{div(}}\phi \prime ({{\left| {\nabla u} \right|}^2})\nabla u{\rm{) + }}{{\left| u \right|}^{\alpha - 2}}u = \lambda h(x,u),} \hfill \cr {u(x) \to 0,\;\;\;\;\;{\rm{as}}\left| x \right| \to \infty ,} \hfill \cr } } \right.\;\;\;\;\;\;\;\;{\rm{in}}\;\;\;\;{\mathbb{R}^N},$$ where N ≥ 2, 1 < p < q < N, α < q, 1 < α < p*q†/p† with $${p^ * } = {\textstyle{{Np} \over {N - p}}},\;\;\phi (t)$$ behaves like tq/2 for small t and tp/2 for large t, and p′ and q′ are the conjugate exponents of p and q, respectively. We study the existence of nontrivial radially symmetric solutions for the problem above by applying the mountain pass theorem and the fountain theorem. Moreover, taking into account the dual fountain theorem, we show that the problem admits a sequence of small-energy, radially symmetric solutions.

GENERALIZED BESSEL AND FRAME MEASURES

Considering a finite Borel measure $ \mu $ on $ \mathbb{R}^d $, a pair of conjugate exponents $ p, q $, and a compatible semi-inner product on $ L^p(\mu) $, we have introduced $ (p,q) $-Bessel and $ (p,q) $-frame measures as a generalization of the concepts of Bessel and frame measures. In addition, we have defined the notions of $ q $-Bessel sequence and $ q$-frame in the semi-inner product space $ L^p(\mu) $. Every finite Borel measure $\nu$ is a $(p,q)$-Bessel measure for a finite measure $ \mu $. We have constructed a large number of examples of finite measures $ \mu $ which admit infinite $ (p,q) $-Bessel measures $ \nu $. We have showed that if $ \nu $ is a $ (p,q) $-Bessel/frame measure for $ \mu $, then $ \nu $ is $ \sigma $-finite and it is not unique. In fact, by using the convolutions of probability measures, one can obtain other $ (p,q) $-Bessel/frame measures for $ \mu $. We have presented a general way of constructing a $ (p,q) $-Bessel/frame measure for a given measure.

A Note on Interpolation in Hardy Spaces

This note deals with an interpolation problem in the disk. We impose that the interpolation be performed exclusively by the first derivative of a function in a certain Hardy space $$H^p$$ . When $$1<p<\infty $$ , we characterize the corresponding interpolating sequences as the separated ones that also verify a condition for all functions in $$H^q$$ (p and q are conjugate exponents). We also prove that the interpolating sequences for $$p=1$$ are the same as for $$p=2$$ .

On Lebesgue Integrability of Fourier Transforms in Amalgam Spaces

Let f be an element of the subspace $$(L^{q},l^{p})^{\alpha }({\mathbb {R}}^d)$$ ( $$1\le q \le \alpha \le p \le 2 $$ ) of the Wiener amalgam space $$(L^{q},l^{p})({\mathbb {R}}^{d})$$ . It is well-known that the Fourier transform $$\widehat{f}$$ of f belongs to $$(L^{p'},l^{q'})^{\alpha '}({\mathbb {R}}^d)$$ where $$p'$$ , $$q'$$ and $$\alpha '$$ are the conjugate exponents of p, q and $$\alpha $$ respectively. We give sufficient conditions, in terms of a modulus of continuity of f, for $$\widehat{f}$$ to be in a Lebesgue space $$L^{\beta }({\mathbb {R}}^{d})$$ or in an amalgam space $$(L^{p'},l^{s})^{\beta }({\mathbb {R}}^d)$$ other than $$(L^{p'},l^{q'})^{\alpha '}({\mathbb {R}}^d)$$ . As an application, we obtain a sufficient condition for the solvability in $$[L^{2}({\mathbb {R}}^d)]^d$$ of the equation $$\hbox {div}{F}=f$$ .

Hilbert inequality on grand function spaces

In this paper, we study the Hilbert inequality with conjugate exponents in the framework of fully measurable grand Lebesgue spaces and grand Bochner Lebesgue spaces.

It\xf4 formula for processes taking values in intersection of finitely many Banach spaces

Motivated by applications to SPDEs we extend the Itô formula for the square of the norm of a semimartingale y(t) from Gyöngy and Krylov (Stochastics 6(3):153–173, 1982) to the case ∑i=1m∫(0,t]vi∗(s)dA(s)+h(t)=:y(t)∈VdA×P-a.e.,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{i=1}^m \\int _{(0,t]} v_i^{*}(s)\\,dA(s) + h(t)=:y(t)\\in V \\quad dA\\times {\\mathbb {P}}\\text {-a.e.}, \\end{aligned}$$\\end{document}where A is an increasing right-continuous adapted process, v_i^{*} is a progressively measurable process with values in V_i^{*}, the dual of a Banach space V_i, h is a cadlag martingale with values in a Hilbert space H, identified with its dual H^{*}, and V:=V_1cap V_2 cap cdots cap V_m is continuously and densely embedded in H. The formula is proved under the condition that Vert yVert _{V_i}^{p_i} and Vert v_i^*Vert _{V_i^*}^{q_i} are almost surely locally integrable with respect to dA for some conjugate exponents p_i, q_i. This condition is essentially weaker than the one which would arise in application of the results in Gyöngy and Krylov (Stochastics 6(3):153–173, 1982) to the semimartingale above.

Radial solutions of quasilinear equations in Orlicz–Sobolev type spaces

This paper is devoted to prove the existence of a nontrivial nonnegative radial solution for the quasilinear elliptic equation−div(φ(|∇u|)∇u)+|u|α−2u=|u|s−2uin RN where N≥2, 1<α≤l⁎m′l′, max⁡{m,α}<s<l⁎, being l,m∈(1,N), l⁎=lNN−l and m′ and l′ the respective conjugate exponents of m and l. The function φ is allowed to belong to a larger class, which includes the special cases appearing in mathematical models in the fields of physics, for instance, nonlinear elasticity, plasticity and generalized Newtonian fluids. The proof is based on variational methods in the Orlicz–Sobolev spaces.

Inequalities with conjugate exponents in grand Lebesque spaces

In this paper we want to show the validity of the generalized doubleparametric Hilbert inequality with conjugate exponents in the framework of grand Lebesgue spaces as a particular case of a more general result involving an homogeneous kernel. We also study the boundedness of an integral operator with the aforementioned kernel. 2000 AMS Classification: Primary 46E30; Secondary 26D15

Sharp [formula omitted] estimates for discrete second order Riesz transforms

We show that multipliers of second order Riesz transforms on products of discrete abelian groups enjoy the Lp estimate p⁎−1, where p⁎=max{p,q} and p and q are conjugate exponents. This estimate is sharp if one considers all multipliers of the form ∑iσiRiRi⁎ with |σi|⩽1 and infinite groups. In the real valued case, we obtain better sharp estimates for some specific multipliers, such as ∑iσiRiRi⁎ with 0⩽σi⩽1. These are the first known precise Lp estimates for discrete Calderón–Zygmund operators.

On reverse Hilbert-type inequalities

By introducing two pairs of conjugate exponents and estimating the weight coefficients, we establish reverse versions of Hilbert-type inequalities, as described by Jin (J. Math. Anal. Appl. 340:932-942, 2008), and we prove that the constant factors are the best possible. As applications, some particular results are considered.

A half-discrete Hilbert-type inequality with the non-monotone kernel and the best constant factor

By introducing two pairs of conjugate exponents and using the improved Euler-Maclaurin summation formula, we estimate the weight functions and obtain a half-discrete Hilbert-type inequality with the non-monotone kernel and the best constant factor. We also consider its equivalent forms.

Functional Inequalities and Hamilton–Jacobi Equations in Geodesic Spaces

We study the connection between the p-Talagrand inequality and the q-logarithmic Sololev inequality for conjugate exponents p ≥ 2, q ≤ 2 in proper geodesic metric spaces. By means of a general Hamilton–Jacobi semigroup we prove that these are equivalent, and moreover equivalent to the hypercontractivity of the Hamilton–Jacobi semigroup. Our results generalize those of Lott and Villani. They can be applied to deduce the p-Talagrand inequality in the sub-Riemannian setting of the Heisenberg group.

Hilbert-type integral inequality with the homogeneous kernel of 0-degree

Through using weight function, we give a new Hilbert-type integral inequality with two independent parameters and two pair of conjugate exponents, which is a best extension of a Hilbert-type integral inequality with the homogeneous kernel of 0-degree. The equivalent form, the reverses and some particular results are considered.

The Hilbert-Type Integral Inequality with the System Kernel of-λ Degree Homogeneous Form

In this paper, the integral operator is used. We give a new Hilbert-type integral inequality, whose kernel is the homogeneous form with degree - and with three pairs of conjugate exponents and the best constant factor and its reverse form are also derived. It is shown that the results of this paper represent an extension as well as some improvements of the earlier results.

On a Multiple Hilbert's Inequality with Parameters

By introducing multiparameters and conjugate exponents and using Hadamard's inequality and the way of real analysis, we estimate the weight coefficients and give a multiple more accurate Hilbert's inequality, which is an extension of some published results. We also prove that the constant factor in the new inequality is the best possible and consider its equivalent form.

Hilbert-type inequalities and related operators with homogeneous kernel of degree 0

In this paper we provide an unified approach to the Hilbert-type inequalities with homogeneous kernel of degree 0 and certain weight functions. As an application, we define the related Hilbert-type operators and analyze their norms. In the case of conjugate exponents, we obtain the best possible constants involved in the right-hand sides of derived inequalities, and norms of the Hilbert-type operators as well. Finally, we consider some special choices of homogeneous kernels and parameters.

A New Dual Hardy-Hilbert's Inequality with some Parameters and its Reverse

By using the improved Euler-Maclaurin summation formula and estimating the weight coefficients in this paper, a new dual Hardy-Hilbert's inequality and its reverse form are obtained, which are all with two pairs of conjugate exponents (p, q); (r, s) and a independent parameter . In addition, some equivalent forms of the inequalities are considered. We also prove that the constant factors in the new inequalities are all the best possible. As a particular case of our results, we obtain the reverse form of a famous Hardy-Hilbert's inequality.

Hardy-Hilbert's inequality with general homogeneous kernel

A general form of recently obtained Hardy-Hilbert’s inequalitywith perturbed Hilbert’s kernel with the best possible estimation in the case of conjugate exponents is obtained. The multidimensional case is also considered. The case of non-conjugate exponents is briefly given. Mathematics subject classification (2000): 26D15.

General Hilbert-type inequalities with non-conjugate exponents

In this paper we derive a series of new one-dimensional and multidimensional integral and discrete inequalities of the Hilbert and the Hardy-Hilbert type, with non-conjugate exponents. First, prove and discuss two equivalent general inequalities of such type, as well as their corresponding reverse inequalities. The obtained results are then applied to various settings considering homogeneous functions of a negative real degree. In particular, we prove generalizations and refinements of some recent results of Rassias et al, related to the Hilbert-type inequalities with conjugate exponents, and some new multidimensional inequalities of the Godunova type. Mathematics subject classification (2000): 26D10, 26D15.