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New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights

100%

Kałamajska A., Pietruska-Pałuba K.

Open Mathematics

2012

tom 10

nr 6

2033-2050

We obtain Hardy type inequalities $$\int_0^\infty {M\left( {\omega \left( r \right)\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr} \leqslant C_1 \int_0^\infty {M\left( {\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr + C_2 \int_0^\infty {M\left( {\left| {u'\left( r \right)} \right|} \right)\rho \left( r \right)dr,} }$$ and their Orlicz-norm counterparts $$\left\| {\omega u} \right\|_{L^M (\mathbb{R}_ + ,\rho )} \leqslant \tilde C_1 \left\| u \right\|_{L^M (\mathbb{R}_ + ,\rho )} + \tilde C_2 \left\| {u'} \right\|_{L^M (\mathbb{R}_ + ,\rho )} ,$$ 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.

Mapping properties of integral averaging operators

88%

P. Heinig H., Sinnamon G.

Studia Mathematica

1998

tom 129

nr 2

157-177

Characterizations are obtained for those pairs of weight functions u and v for which the operators $Tf(x) = ʃ_{a(x)}^{b(x)} f(t)dt$ with a and b certain non-negative functions are bounded from $L^p_u(0,∞)$ to $L^q_v(0,∞)$, 0 < p,q < ∞, p≥ 1. Sufficient conditions are given for T to be bounded on the cones of monotone functions. The results are applied to give a weighted inequality comparing differences and derivatives as well as a weight characterization for the Steklov operator.

Ograniczanie wyników

1 Open Mathematics

1 Studia Mathematica

1 Kałamajska A.

1 P. Heinig H.

1 Pietruska-Pałuba K.

1 Sinnamon G.

1 2012

1 1998

Wyniki wyszukiwania

New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights

Mapping properties of integral averaging operators