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1.
We prove a sharp estimate for the k-modulus of smoothness, modelled upon a -Lebesgue space, of a function f in , where Ω is a domain with minimally smooth boundary and finite Lebesgue measure, , and . This sharp estimate is used to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces into generalized Hölder spaces defined by means of the k-modulus of smoothness. General results are illustrated with examples. In particular, we obtain a generalization of the classical Jawerth embeddings. 相似文献
2.
Amiran Gogatishvili Rza Mustafayev Tuğçe Ünver 《Czechoslovak Mathematical Journal》2017,67(4):1105-1132
In this paper, characterizations of the embeddings between weighted Copson function spaces \(Co{p_{{p_1},{q_1}}}\left( {{u_1},{v_1}} \right)\) and weighted Cesàro function spaces \(Ce{s_{{p_2},{q_2}}}\left( {{u_2},{v_2}} \right)\) are given. In particular, two-sided estimates of the optimal constant c in the inequality where p1, p2, q1, q2 ∈ (0,∞), p2 ≤ q2 and u1, u2, v1, v2 are weights on (0,∞), are obtained. The most innovative part consists of the fact that possibly different parameters p1 and p2 and possibly different inner weights v1 and v2 are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Cesàro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.
相似文献
$${\left( {\int_0^\infty {{{\left( {\int_0^t {f{{\left( \tau \right)}^{{p_2}}}{v_2}\left( \tau \right)d\tau } } \right)}^{{q_2}/{p_2}}}{u_2}\left( t \right)dt} } \right)^{1/{q_2}}} \leqslant c{\left( {\int_0^\infty {{{\left( {\int_t^\infty {f{{\left( \tau \right)}^{{p_1}}}{v_1}\left( \tau \right)d\tau } } \right)}^{{q_1}/{p_1}}}{u_1}\left( t \right)dt} } \right)^{1/{q_1}}},$$
3.
We establish necessary and sufficient conditions for embeddings of Bessel potential spaces H σ X(IR n ) with order of smoothness σ?∈?(0, n), modelled upon rearrangement invariant Banach function spaces X(IR n ), into generalized Hölder spaces (involving k-modulus of smoothness). We apply our results to the case when X(IR n ) is the Lorentz-Karamata space \(L_{p,q;b}({{\rm I\kern-.17em R}}^n)\). In particular, we are able to characterize optimal embeddings of Bessel potential spaces \(H^{\sigma}L_{p,q;b}({{\rm I\kern-.17em R}}^n)\) into generalized Hölder spaces. Applications cover both superlimiting and limiting cases. We also show that our results yield new and sharp embeddings of Sobolev-Orlicz spaces W k?+?1 L n/k(logL) α (IR n ) and W k L n/k(logL) α (IR n ) into generalized Hölder spaces. 相似文献
4.
If ${p(\cdot): \mathbb{R}^{n} {\rightarrow} (0,\infty), n\geq 3}$ , is globally log-Hölder continuous and its infimum p ? and its supremum p + are such that ${\frac{n}{n-1} < p^{-} \leq p^{+} < p^{-} (n-1)}$ , then the spherical maximal operator (integral averages taken with respect to the (n ? 1)-dimensional surface measure) is bounded. When n = 3, the result is then interpreted as the preservation of the integrability properties of the initial velocity of propagation to the solution of the initial-value problem for the wave equation. 相似文献
5.
We study the question of convergence of Padé and Padé-type approximants to functions meromorphic in a domain. As an example we investigate in detail the case of functions of the formwhereμ(z) is a Markov function. 相似文献
6.
We investigate the following problem: For which open simply connected domains do there exist interpolation schemes (a set of interpolation points) such that for any analytic function defined in the domain the corresponding interpolating polynomials converge to the function when the degree of the polynomials tends to infinity? We also study similar problems for rational interpolants. These problems are connected to the balayage (sweeping out) problems of measures. 相似文献
7.
Amiran Gogatishvili Lars‐Erik Persson Vladimir D. Stepanov Peter Wall 《Mathematische Nachrichten》2014,287(2-3):242-253
We prove that the weighted Stieltjes inequality can be characterized by four different scales of conditions also for the case , . In particular, a new proof of a result of G. Sinnamon 10 is given, which also covers the case . Moreover, for the situation at hand a new gluing theorem and also an equivalence theorem of independent interest are proved and discussed. By applying our new equivalence theorem to weighted inequalities for the Stieltjes transform we obtain the four new scales of conditions for characterization of Stieltjes inequality. 相似文献
8.
Edoh Y. Amiran 《Communications in Mathematical Physics》1993,154(1):99-110
The Lazutkin parameter for curves which are invariant under the billiard ball map is viewed symplectically in a way which makes it analogous to the sum of the values of a generating function over a closed orbit. This leads to relations among lengths of closed geodesics, lengths of invariant curves for the billiard map, rotation numbers, and the Lazutkin parameter. These relations establish the Birkhoff invariant and the expansion for the lengths of invariant curves in terms of the Lazutkin parameter as symplectic and spectral invariants (for the Dirichlet spectrum) and provide invariants which characterize a family of ellipses among smooth curves with positive curvature. Geodesic flow on a bounded planar region gives rise to several geometric objects among which are closed reflected geodesics and invariant curves-closed curves whose tangents are invariant under reflection at the boundary. On a bounded domain, the map that assigns to each geodesic segment its successor after reflection at the boundary is called the billiard ball map and its dual (in the cotangent bundle for the boundary) is called the boundary map. 相似文献
9.
10.
Antnio M. Caetano Amiran Gogatishvili Bohumir Opic 《Journal of Approximation Theory》2008,152(2):188-214
We use Kolyada's inequality and its converse form to prove sharp embeddings of Besov spaces (involving the zero classical smoothness and a logarithmic smoothness with the exponent β) into Lorentz–Zygmund spaces. We also determine growth envelopes of spaces . In distinction to the case when the classical smoothness is positive, we show that we cannot describe all embeddings in question in terms of growth envelopes. 相似文献