General Tauberian theorems for the Cesàro integrability of functions

For a locally integrable function f on {[0,\infty)}, we defineF(t)=\int_{0}^{t}f(u)\mathop{}\!du\quad\text{and}\quad\sigma_{\alpha}(t)=\int_% {0}^{t}\biggl{(}1-\frac{u}{t}\biggr{)}^{\alpha}f(u)\mathop{}\!dufor {t>0}. The improper integral {\int_{0}^{\infty}f(u)\mathop{}\!du} is said to be {(C,\alpha)} integrable to L for some {\alpha>-1} if the limit {\lim_{x\to\infty}\sigma_{\alpha}(t)=L} exists. It is known that {\lim_{t\to\infty}F(t)=\ell} implies {\lim_{t\to\infty}\sigma_{\alpha}(t)=\ell} for {\alpha>-1}, but the converse of this implication is not true in general. In this paper, we introduce the concept of the general control modulo of non-integer order for functions and obtain some Tauberian conditions in terms of this concept for the {(C,\alpha)} integrability method in order that the converse implication hold true. Our results extend the main theorems in [Ü. Totur and İ. Çanak, Tauberian conditions for the (C,\alpha) integrability of functions, Positivity 21 2017, 1, 73–83].

A note on the trace inequality for Riesz potentials

We establish a necessary and sufficient condition on a non-negative locally integrable function v guaranteeing the (trace) inequality\lVert I_{\alpha}f\rVert_{L^{p}_{v}(\mathbb{R}^{n})}\leq C\lVert f\rVert_{L^{p% ,1}(\mathbb{R}^{n})}for the Riesz potential {I_{\alpha}}, where {L^{p,1}(\mathbb{R}^{n})} is the Lorentz space. The same problem is studied for potentials defined on spaces of homogeneous type.

Sobolev mappings and moduli inequalities on Carnot groups

We study the mappings that satisfy moduli inequalities on Carnot groups. We prove that the homeomorphisms satisfying the moduli inequalities ($Q$-homeomor\-phisms) with a locally integrable function $Q$ are Sobolev mappings. On this base in the frameworks of the weak inverse mapping theorem, we prove that, on the Carnot groups $\mathbb G,$ the mappings inverse to Sobolev homeomorphisms of finite distortion of the class $W^1_{\nu,\loc}(\Omega;\Omega')$ belong to the Sobolev class $W^1_{1,\loc}(\Omega';\Omega)$.

Criteria of Removable Sets for Harmonic Functions in the Sobolev Spaces L1p,w

Ahlfors and Beurling [16] proved that set 𝐸 is removable for class 𝐴𝐷2 of analytic functions with the finite Dirichlet integral if and only if 𝐸 does not change extremal distances. Their proof uses the conformal invariance of class 𝐴𝐷2, so it does not immediately generalize to 𝑝 ̸= 2 and to the relevant classes of harmonic functions in the space. In 1974 Hedberg [19] proposed new approaches to the problem of describing removable singularities in the function theory. In particular he gave the exact functional capacitive conditions for a set to be removable for class 𝐻𝐷𝑝(𝐺). Here 𝐻𝐷𝑝(𝐺) is the class of real-valued harmonic functions 𝑢 in a bounded open set 𝐺 ⊂ 𝑅𝑛, 𝑛 ≥ 2, and such that ∫︁ 𝐺 |∇𝑢|𝑝 𝑑𝑥 < ∞, 𝑝 > 1. In this paper we extend Hedberg’s results on class 𝐻𝐷𝑝,𝑤(𝐺) of harmonic functions 𝑢 in 𝐺 and such that ∫︁ 𝐺 |∇𝑢|𝑝 𝑤𝑑𝑥 < ∞. Here a locally integrable function 𝑤 : 𝑅𝑛 → (0,+∞) satisfies the Muckenhoupt condition [20] sup 1 |𝑄| ∫︁ 𝑄 𝑤𝑑𝑥 ⎛ ⎝ 1 |𝑄| ∫︁ 𝑄 𝑤1−𝑞𝑑𝑥 ⎞ ⎠ 𝑝−1 < ∞, where the supremum is taking over all coordinate cubes 𝑄 ⊂ 𝑅𝑛, 𝑞 ∈ (1,+∞) and 1 𝑝 + 1 𝑞 = 1; by ℒ𝑛(𝑄) = |𝑄| we denote the 𝑛-dimensional Lebesgue measure of 𝑄. We denote by 𝐿1 𝑞 , ˜ 𝑤(𝐺) the Sobolev space of locally integrable functions 𝐹 on 𝐺, whose generalized gradient in 𝐺 are such that ‖𝑓‖𝐿1 𝑞 , ˜ 𝑤(𝐺) = ⎛ ⎝ ∫︁ 𝐺 |∇𝑓|𝑞 ˜ 𝑤𝑑𝑥 ⎞ ⎠ 1 𝑞 < ∞, where ˜ 𝑤 = 𝑤1−𝑞. The closure of 𝐶∞ 0 (𝐺) in ‖ · ‖𝐿1 𝑞 , ˜ 𝑤(𝐺) is denoted by ∘L 1 𝑞, ˜ 𝑤(𝐺). For compact set 𝐾 ⊂ 𝐺 (𝑞, ˜ 𝑤)-capacity regarding 𝐺 is defined by 𝐶𝑞, ˜ 𝑤(𝐾) = inf 𝑣 ∫︁ 𝐺 |∇𝑣|𝑞 ˜ 𝑤𝑑𝑥, where the infimum is taken over all 𝑣 ∈ 𝐶∞ 0 (𝐺) such that 𝑣 = 1 in some neighbourhood of 𝐾. Note that 𝐶𝑞, ˜ 𝑤(𝐾) = 0 is independent from the choice of bounded set 𝐺 ⊂ 𝑅𝑛. We set 𝐶𝑞, ˜ 𝑤(𝐹) = 0 for arbitrary 𝐹 ⊂ 𝑅𝑛 if for every compact 𝐾 ⊂ 𝐹 there exists a bounded open set 𝐺 such that 𝐶𝑞, ˜ 𝑤(𝐾) = 0 regarding 𝐺. To conclude, we formulate the main results. Theorem 1. Compact 𝐸 ⊂ 𝐺 is removable for 𝐻𝐷𝑝,𝑤(𝐺) if and only if 𝐶∞ 0 (𝐺 ∖ 𝐸) is dense in ∘L 1 𝑞, ˜ 𝑤(𝐺). Theorem 2. Compact 𝐸 ⊂ 𝐺 is removable for 𝐻𝐷𝑝,𝑤(𝐺) if and only if 𝐶𝑞, ˜ 𝑤(𝐸) = 0. Corollary. The property of being removable for 𝐻𝐷𝑝,𝑤(𝐺) is local, i.e. compact 𝐸 ⊂ 𝐺 is removable if and only if every 𝑥 ∈ 𝐸 has a compact neighbourhood, whose intersection with 𝐺 is removable. Theorem 3. If 𝐺 is an open set in 𝑅𝑛 and 𝐶𝑞, ˜ 𝑤(𝑅𝑛 ∖𝐺) = 0. Then 𝐶∞ 0 (𝐺) is dense in ∘L 1 𝑞, ˜ 𝑤(𝑅𝑛).

Bilinear Multipliers on Banach Function Spaces

Let X1,X2,X3 be Banach spaces of measurable functions in L0(R) and let m(ξ,η) be a locally integrable function in R2. We say that m∈BM(X1,X2,X3)(R) if Bm(f,g)(x)=∫R∫Rf^(ξ)g^(η)m(ξ,η)e2πi<ξ+η,x>dξdη, defined for f and g with compactly supported Fourier transform, extends to a bounded bilinear operator from X1×X2 to X3. In this paper we investigate some properties of the class BM(X1,X2,X3)(R) for general spaces which are invariant under translation, modulation, and dilation, analyzing also the particular case of r.i. Banach function spaces. We shall give some examples in this class and some procedures to generate new bilinear multipliers. We shall focus on the case m(ξ,η)=M(ξ-η) and find conditions for these classes to contain nonzero multipliers in terms of the Boyd indices for the spaces.

CHARACTERIZATIONS OF BMO AND LIPSCHITZ SPACES IN TERMS OF WEIGHTS AND THEIR APPLICATIONS

Let $0<\unicode[STIX]{x1D6FC}<n,1\leq p<q<\infty$ with $1/p-1/q=\unicode[STIX]{x1D6FC}/n$, $\unicode[STIX]{x1D714}\in A_{p,q}$, $\unicode[STIX]{x1D708}\in A_{\infty }$ and let $f$ be a locally integrable function. In this paper, it is proved that $f$ is in bounded mean oscillation $\mathit{BMO}$ space if and only if $$\begin{eqnarray}\sup _{B}\frac{|B|^{\unicode[STIX]{x1D6FC}/n}}{\unicode[STIX]{x1D714}^{p}(B)^{1/p}}\bigg(\int _{B}|f(x)-f_{\unicode[STIX]{x1D708},B}|^{q}\unicode[STIX]{x1D714}(x)^{q}\,dx\bigg)^{1/q}<\infty ,\end{eqnarray}$$ where $\unicode[STIX]{x1D714}^{p}(B)=\int _{B}\unicode[STIX]{x1D714}(x)^{p}\,dx$ and $f_{\unicode[STIX]{x1D708},B}=(1/\unicode[STIX]{x1D708}(B))\int _{B}f(y)\unicode[STIX]{x1D708}(y)\,dy$. We also show that $f$ belongs to Lipschitz space $Lip_{\unicode[STIX]{x1D6FC}}$ if and only if $$\begin{eqnarray}\sup _{B}\frac{1}{\unicode[STIX]{x1D714}^{p}(B)^{1/p}}\bigg(\int _{B}|f(x)-f_{\unicode[STIX]{x1D708},B}|^{q}\unicode[STIX]{x1D714}(x)^{q}\,dx\bigg)^{1/q}<\infty .\end{eqnarray}$$ As applications, we characterize these spaces by the boundedness of commutators of some operators on weighted Lebesgue spaces.

Local k-convoluted c-semigroups and complete second order abstract Cauchy problems

Let C : X ? X be a bounded linear operator on a Banach space X over the field F(=R or C), and K : [0,T0)?F a locally integrable function for some 0 < T0 ? ?. Under some suitable assumptions, we deduce some relationship between the generation of a local (or an exponentially bounded) K-convoluted (C 0 0 C)-semigroup on X x X with subgenerator (resp., the generator) (0 I B A) and one of the following cases: (i) the well-posedness of a complete second-order abstract Cauchy problem ACP(A,B,f,x,y): w??(t) = Aw?(t) + Bw(t) + f (t) for a.e. t ?(0,T0) with w(0) = x and w?(0) = y; (ii) a Miyadera-Feller-Phillips-Hille- Yosida type condition; (iii) B is a subgenerator (resp., the generator) of a locally Lipschitz continuous local ?-times integrated C-cosine function on X for which A may not be bounded; (iv) A is a subgenerator (resp., the generator) of a local ?-times integrated C-semigroup on X for which B may not be bounded.

Local K-convoluted C-cosine functions and abstract cauchy problems

Let K : [0,T0)? F be a locally integrable function, and C : X ? X a bounded linear operator on a Banach space X over the field F(=R or C). In this paper, we will deduce some basic properties of a nondegenerate local K-convoluted C-cosine function on X and some generation theorems of local Kconvoluted C-cosine functions on X with or without the nondegeneracy, which can be applied to obtain some equivalence relations between the generation of a nondegenerate local K-convoluted C-cosine function on X with subgenerator A and the unique existence of solutions of the abstract Cauchy problem: U''(t)=Au(t)+f(t) for a.e. t ? (0, T0), u(0) = x, u'(0) = y when K is a kernel on [0, T0), C : X ? X an injection, and A : D(A) ? X ? X a closed linear operator in X such that CA ? AC. Here 0 < T0 ? ?, x,y ? X, and f ? L1,loc([0,T0),X).

Local Integral Estimates for Quasilinear Equations with Measure Data

Local integral estimates as well as local nonexistence results for a class of quasilinear equations-Δpu=σP(u)+ωforp>1and Hessian equationsFk-u=σP(u)+ωwere established, whereσis a nonnegative locally integrable function or, more generally, a locally finite measure,ωis a positive Radon measure, andP(u)~exp⁡αuβwithα>0andβ≥1orP(u)=up-1.