We present two new general \(L^p\)-integral inequalities involving the primitive of a function. They extend the applicability of some established results in the literature, mainly thanks to the presence of an intermediate function that can be adapted to the context. We thus contribute to the understanding of the interaction between the \(L^p\)-integrals of functions and their primitives.
We survey recent developments in the theory of achievement sets and present a substantial collection of open problems.
Let U be an open set in ℝ^d. A continuous function f U →ℝ is strongly nowhere differentiable if and only if for each γ∈(0,1] and for each unit speed C^1,γ curve c [a,b] → U, the composition f∘ c [a,b] →ℝ is nowhere differentiable on (a,b). For bounded U, let U be the closure of U and C(U) be the Banach space of continuous real-valued functions on U with the sup norm. Theorem. In the sense of the Baire category theorem, almost every f∈ C(U) is strongly nowhere differentiable on U.
We present a new type of integral that is supposed to extend the usability of the Lebesgue integral in certain types of investigations. It is based on the Hausdorff dimension and measure. We examine the basic properties of the integral and its similarities to the properties of the Lebesgue integral. We present many applications as well.
In this paper we will deal with problems in approximation theory of bounded analytic functions on the unit disc and their boundary behavior on the unit circle. We will attempt to unify two known such theorems to create a stronger theorem. We will solve various special cases of the theorem, and in particular one case extends the main result found in \cite{PE} regarding the boundary behavior of Blaschke products. The necessity part of the proof uses a classical theorem of Baire. We will see that the proof of the necessity part of the extension theorem provides a simplification and a more elegant approach for the necessity part of the main result found in \cite{PE}. Lastly, we will prove an analogue of a classical theorem by Kolesnikov for Blaschke products provided an extra requirement is satisfied and use the main result in \cite{PE} to simplify the proof of a result found in \cite{PI}.
A contraction in a complete metric space is a Picard operator but a contractive map in a complete metric space need not even have a fixed point. We introduce the notion of a Picard space. It is a metric space in which every contractive map having a fixed point is a Picard operator. We then find an equivalent definition of a Picard space which allows us to show that every Heine-Borel space is Picard. Finally, we describe the Venn diagram for the classes of complete, Picard and Heine-Borel spaces.
We introduce the concept of [rho(1), rho(2)]-upper superdense sets and study their properties. These are measurable sets E such that, for every x is an element of E, the intersection of E with any set of density not less than rho 2 has density not less than g1. We investigate nontrivial examples, establish basic properties of [rho 1, rho 2]-upper superdense sets, and analyze the relationships between these families for different values of rho 1, rho 2. The families of these sets are a generalization of both the density topology and the T & lowast; topology. Namely, all the discussed families contain the density topology as a subset, while the T & lowast; topology is exactly the family of [1, 1]-superdense sets. Furthermore, we demonstrate how the notion of [rho 1, rho 2]-upper superdensity can be applied to adders of g-upper continuous functions.
We construct a multiplicative group of cardinality continuum that is contained (with the exception of the constant function equal to one) in the class of Hamel functions and prove that there is no such group of greater cardinality. Additionally, we answer in the negative [4, Question 1] whether there exists a free group of the size greater than c with respect to the operation of composition that is contained (with the exception of the identity function) in the class of Hamel functions. We also prove the nonexistence of such large free groups with respect to other operations on functions (addition, multiplication).
This note constructs a two-times totally differentiable function such that the second derivative is discontinuous but the trace of the second derivative is continuous. Thus there exists bounded and continuous data such that the Poisson equation can be satisfied pointwise everywhere by a solution that is not classical.
There are two main contributions in this paper. First, the Xiw-sum of Banach spaces is defined, following the work of Argyros and Motakis in [4]. In reality, this Banach space is an example of a U-sum of Banach spaces (e.g. in [12]), however, the alternate construction that is provided in this paper allows for a self-contained proof that it has the Lebesgue property if each summand has the Lebesgue property. This second result not only obtains a large new class of Banach spaces with the Lebesgue property, but it also shows that the non-asymptotic-P1 sum of infinitedimensional Banach spaces still can have the Lebesgue property, which had not been established previously.
A notion of Lr-delta-variation of a function in Lr which plays an essential role in the theory of the Henstock-Kurzweil integral for functions in Lr (HKr-integral) is studied. We obtain an improved, in fact the best possible, estimate from below for this variation via the classical variation. We show that the class of functions having a finite Lr-delta-variation on an interval coincides with the class of functions of bounded variation on this interval. As a by-product of the results of the paper we obtain a new proof of the uniqueness of the HKr-integral.
Consider X as a complete separable metric space. This paper demonstrates that, for a typical probability measure, the general lower local dimension is consistently zero, while the general upper local dimension remains infinite. Additionally, the general lower local dimension is almost always zero, and given certain additional conditions on X, a corresponding outcome is established for the general upper local dimension. Furthermore, we establish that the overall Hausdorff dimension of a typical measure on the space X is consistently zero within any compact space. Simultaneously, under specific additional conditions, we show that the general packing dimension surpasses a predefined value s > 0. More precisely, we show that the local dimension of a typical measure fails to exist. Specifically, the behavior of a typical measure nu is so extremely irregular that, for a fixed point x is an element of X, the generalized local dimension function r bar right arrow phi(nu(B-r(x)))/psi(r), of v at x, with respect to the functions phi and psi, remains divergent as r -> 0, even after being "averaged" or "smoothed out" by general and powerful averaging methods. These include, for example, higher-order Riesz-Hardy logarithmic averages and Cesaro averages.
Let X be a separable Banach space with the dual space X* and 1 < p < infinity. We study some properties of p-frames for X having the form {T-n phi}(n is an element of I) where either I = N boolean OR {0} or I = Z and T : X* -> X* is a bounded linear operator. As a main result, for a p-frame {T-phi(k)}(k is an element of Z), we determine the structure of the canonical dual and also characterize the operator of any dual having the same construction. In the case I = N boolean OR {0}, for a p-frame {Tn phi}infinity n=0, we get a stability result by considering perturbations of the vector phi is an element of X* from an invariant subspace for T. Finally, based on the concept of epsilon-approximation of a p-frame in Banach spaces, we will approximate any p-frame {gi}(infinity)(i=1) via a sub-orbit of a bounded linear operator T : X*-> X* such as hypercyclic operators. This result creates a large class of p-frames generated by operator actions.
For certain Banach spaces, we establish the lineability of the set of Riemann integrable functions which are not Darboux integrable.
We show that the product of density differentiation bases B1,. . . , Bk is a density basis as well, provided the factor bases Bj are measurable in the sense that the values of the truncated maximal operators associated with them are measurable for characteristic functions of finite unions of intervals.
We give sufficient conditions for some convolution operators, which include Laplace transform, Gauss-Weierstrass and Poisson operators, one-dimensional Riesz potentials, and Riemann-Liouville and Weyl fractional integrals, to verify weighted mixed weak-type inequalities.
