scholarly journals ON SOME REFINEMENTS AND CONVERSES OF MULTIDIMENSIONAL HILBERT-TYPE INEQUALITIES

2012 ◽  
Vol 85 (3) ◽  
pp. 380-394 ◽  
Author(s):  
MARIO KRNIĆ
Keyword(s):  
Type Inequality ◽  
Hölder Inequality ◽  
Weighted Functions ◽  
Difference Form ◽  
Hilbert Type

AbstractThe main objective of this paper is a study of some new refinements and converses of multidimensional Hilbert-type inequalities with nonconjugate exponents. Such extensions are deduced with the help of some remarkable improvements of the well-known Hölder inequality. First, we obtain refinements and converses of the general multidimensional Hilbert-type inequality in both quotient and difference form. We then apply the results to homogeneous kernels with negative degree of homogeneity. Finally, we consider some particular settings with homogeneous kernels and weighted functions, and compare our results with those in the literature.

On a strengthened multidimensional Hilbert-type inequality

2014 ◽  
Vol 64 (5) ◽  
Author(s):  
Mario Krnić
Keyword(s):  
Hölder’S Inequality ◽  
Type Inequality ◽  
Hölder's Inequality ◽  
Weighted Functions ◽  
Hilbert Type

AbstractThe main objective of this paper is a study of the general refinement and converse of the multidimensional Hilbert-type inequality in the so-called quotient form. Such extensions are deduced with the help of the sophisticated use of the well-known Hölder’s inequality. The obtained results are then applied to homogeneous kernels with the negative degree of homogeneity. Also, we establish the conditions under which the constant factors involved in the established inequalities are the best possible. Finally, we consider some particular settings with homogeneous kernels and weighted functions. In such a way we obtain both refinements and converses of some actual results, known from the literature.

scholarly journals On a more accurate Hilbert-type inequality in the whole plane with the general homogeneous kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Xingshou Huang ◽  
Bicheng Yang
Keyword(s):  
Equivalent Form ◽  
Type Inequality ◽  
Constant Factor ◽  
Real Analysis ◽  
Homogeneous Kernel ◽  
Hilbert’S Inequality ◽  
The Best Possible Constant ◽  
Hilbert Type ◽  
Weight Coefficients

AbstractBy the use of the weight coefficients, the idea of introduced parameters and the technique of real analysis, a more accurate Hilbert-type inequality in the whole plane with the general homogeneous kernel is given, which is an extension of the more accurate Hardy–Hilbert’s inequality. An equivalent form is obtained. The equivalent statements of the best possible constant factor related to several parameters, the operator expressions and a few particular cases are considered.

scholarly journals New Generalizations and Results in Shift-Invariant Subspaces of Mixed-Norm Lebesgue Spaces \({L_{\vec{p}}(\mathbb{R}^d)}\)

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 227 ◽  
Author(s):  
Junjian Zhao ◽  
Wei-Shih Du ◽  
Yasong Chen
Keyword(s):  
Invariant Subspace ◽  
Type Inequality ◽  
Invariant Subspaces ◽  
Minkowski Inequality ◽  
Hölder Inequality ◽  
Stability Theorem ◽  
Mixed Norm ◽  
Lebesgue Spaces

In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces Lp→(Rd). We obtain a mixed-norm Hölder inequality, a mixed-norm Minkowski inequality, a mixed-norm convolution inequality, a convolution-Hölder type inequality and a stability theorem to mixed-norm case in the setting of shift-invariant subspace of Lp→(Rd). Our new results unify and refine the existing results in the literature.

scholarly journals A NEW HALF-DISCRETE HILBERT-TYPE INEQUALITY IN THE WHOLE PLANE

2017 ◽  
Vol 7 (3) ◽  
pp. 977-991
Author(s):  
Bicheng Yang ◽  
◽  
Bing He
Keyword(s):  
Type Inequality ◽  
Hilbert Type

scholarly journals INEQUALITIES OF THE HILBERT TYPE IN $\mathbb{R}^{n}$ WITH NON-CONJUGATE EXPONENTS

2008 ◽  
Vol 51 (1) ◽  
pp. 11-26
Author(s):  
Aleksandra Čižmešija ◽  
Ivan Perić ◽  
Predrag Vuković
Keyword(s):  
Sobolev Inequality ◽  
Integral Formula ◽  
Type Inequality ◽  
Upper Bounds ◽  
Conjugate Exponents ◽  
Hilbert Type

AbstractIn this paper we state and prove a new general Hilbert-type inequality in $\mathbb{R}^{n}$ with $k\geq2$ non-conjugate exponents. Using Selberg's integral formula, this result is then applied to obtain explicit upper bounds for the doubly weighted Hardy–Littlewood–Sobolev inequality and some further Hilbert-type inequalities for $k$ non-negative functions and non-conjugate exponents.

scholarly journals A New Hilbert-Type Inequality with Positive Homogeneous Kernel and Its Equivalent Forms

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 342 ◽  
Author(s):  
Bicheng Yang ◽  
Shanhe Wu ◽  
Aizhen Wang
Keyword(s):  
Type Inequality ◽  
Homogeneous Kernel ◽  
Hilbert Type

We establish a new inequality of Hilbert-type containing positive homogeneous kernel ( min { m , n } ) λ and derive its equivalent forms. Based on the obtained Hilbert-type inequality, we discuss its equivalent forms and give the operator expressions in some particular cases.

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