Let 1 < p < ∞, q = p/(p-1) and for $f\in L^p(0,\infty )$ define $F\left(x\right)=(1/x){ʃ}_0^{x}f\left(t\right)dt$, x > 0. Moser’s Inequality states that there is a constant $C_p$ such that $su{p}_{a\le 1}su{p}_{f\in B_p}{ʃ}_0^{\infty }exp[a{x}^q{\left|F\left(x\right)\right|}^q-x]dx=C_p$ where $B_p$ is the unit ball of $L^p$. Moreover, the value a = 1 is sharp. We observe that $F=K_1$ f where the integral operator $K_1$ has a simple kernel K. We consider the question of for what kernels K(t,x), 0 ≤ t, x < ∞, this result can be extended, and proceed to discuss this when K is non-negative and homogeneous of degree -1. A sufficient condition on K is found for the analogue...

In this paper we generalize Opial inequalities in the multidimensional case over balls. The inequalities carry weights and are proved to be sharp. The functions under consideration vanish at the center of the ball.

We consider a Hardy-type inequality with Oinarov's kernel in weighted Lebesgue spaces. We give new equivalent conditions for satisfying the inequality, and provide lower and upper estimates for its best constant. The findings are crucial in the study of oscillation and non-oscillation properties of differential equation solutions, as well as spectral properties.

We obtain Hardy type inequalities $${\int }_0^{\infty }M\left(\omega \left(r\right)\left|u\left(r\right)\right|\right)\rho \left(r\right)dr⩽C_1{\int }_0^{\infty }M\left(\left|u\left(r\right)\right|\right)\rho \left(r\right)dr+C_2{\int }_0^{\infty }M\left(\left|u^{\text{'}}\left(r\right)\right|\right)\rho \left(r\right)dr,$$ and their Orlicz-norm counterparts $${∥\omega u∥}_{L^M({ℝ}_{+},\rho )}⩽{\tilde{C}}_1{∥u∥}_{L^M({ℝ}_{+},\rho )}+{\tilde{C}}_2{∥u^{\text{'}}∥}_{L^M({ℝ}_{+},\rho )},$$ with an N-function M, power, power-logarithmic and power-exponential weights ω, ρ, holding on suitable dilation invariant supersets of C 0∞(ℝ+). Maximal sets of admissible functions u are described. This paper is based on authors’ earlier abstract results and applies them to particular classes of weights.

Let G be a real connected Lie group with polynomial volume growth endowed with its Haar measuredx. Given a C² positive bounded integrable function M on G, we give a sufficient condition for an L² Poincaré inequality with respect to the measure M(x)dx to hold on G. We then establish a nonlocal Poincaré inequality on G with respect to M(x)dx. We also give analogous Poincaré inequalities on Riemannian manifolds and deal with the case of Hardy inequalities.

We revisit the properties of Bessel-Riesz operators and present a different proof of the boundedness of these operators on generalized Morrey spaces. We also obtain an estimate for the norm of these operators on generalized Morrey spaces in terms of the norm of their kernels on an associated Morrey space. As a consequence of our results, we reprove the boundedness of fractional integral operators on generalized Morrey spaces, especially of exponent $1$, and obtain a new estimate for their norm.

Hardy inequalities for the Hardy-type operators are characterized in the amalgam space which involves Banach function space and sequence space.