We show that any equicontractive, self-similar measure arising from the IFS of contractions \((S_{j})\), with self-similar set \([0,1]\), admits an isolated point in its set of local dimensions provided the images of \(S_{j}(0,1)\) (suitably) overlap and the minimal probability is associated with one (resp., both) of the endpoint contractions. Examples include \(m\)-fold convolution products of Bernoulli convolutions or Cantor measures with contraction factor exceeding \(1/(m+1)\) in the biased case and \(1/m\) in the unbiased case. We also obtain upper and lower bounds on the set of local dimensions for various Bernoulli convolutions.

We describe the Bellman function technique for proving sharp inequalities in harmonic analysis. To provide an example along with historical context, we present how it was originally used by Donald Burkholder to prove \(L^p\) boundedness of the \(\pm 1\) martingale transform. Finally, with Burkholder’s result as a blueprint, we use the Bellman function to prove a new result related to the Chang-Wilson-Wolff Inequality.

A relation between Riemann summability and Riemann derivative is established and necessary and sufficient conditions for Riemann summability of trigonometric series are obtained

In this paper, we establish a new fractional identity involving a functions of two independent variables, and then we derive some fractional Hermite-Hadamard type integral inequalities for functions whose modulus of the mixed derivatives are co-ordinated extended \( \left( s_{1},m_{1} \right) \)-\( \left( s_{2},m_{2}\right) \)-preinvex.

We prove sandwich, Hahn-Banach, Fenchel duality theorems and a version of the Moreau-Rockafellar formula for invariant partially ordered vector space-valued operators. As consequences and applications, we give some versions of Farkas and Kuhn-Tucker-type optimization results and separation theorems, we prove the equivalence of these results and give a further application to Tarski-type theorems and probability measures defined on suitable product spaces.

A classical theorem of W.Sierpiński, S. Mazurkiewicz and S.Kempisty says that the class of all differences of lower semicontinuous functions is uniformly dense in the space of all Baire-one functions. We show a generalization of this result to the case when finely continuous functions of either density topologies or both linear and nonlinear potential theory are involved. Moreover, we examine which topological properties play a crucial role when deriving approximation theorems in more general situations.

We present a novel application of Fourier analysis for solving PDEs which is much faster than the usual separation of variables method and, moreover, it implies uniqueness of the obtained solution at the same time.

We study the space of vector valued multipliers of strongly Henstock-Kurzweil \((\mathcal{SHK})\) integrable functions. We prove that if \(X\) is a commutative Banach algebra, with identity \(e\) of norm one, satisfying Radon-Nikodym property and \(g:[a,b] \rightarrow X\) is of strong bounded variation, then the multiplication operator defined by \(M_g(f)=fg\) maps \(\mathcal{SHK}\) to \(\mathcal{SHK}.\) We also investigate the problems when the domain is \(\mathcal{HK}\) or when \(X\) satisfies weak Radon-Nikodym property.

We study various real-variable techniques for determining whether a function is differentiable without actually calculating a derivative. These include:

• approximation theory \item Fourier analysis

• Sobolev spaces \item Poisson integral

• finite differences

• Campanato-Morrey theory

• Landau's inequalities

In this paper, we introduced the backwards derivative of a Hilbert space-valued function and formulate a version of Fundamental Theorem for the backwards Itô-Henstock integral of an operator-valued stochastic process with respect to a Hilbert space-valued Wiener process.

Sondow and co-workers have employed a key change of variables in order to evaluate double integrals over the unit square \([0,1] \times [0,1]\) in exact closed-form. Motivated by their results, I introduce here a change of variables which creates a ‘bridge’ between integrals of the form \(\,\int_0^1\!\!\int_0^1{f(x \cdot y)~dx \, dy}\,\) and single integrals of the form \(\int_0^1{f(p)\,\ln{p}~d p}\). This allows for prompt closed-form evaluations of several interesting integrals, including some of those investigated recently by Sampedro. I also show that the bridge holds when the intervals of integration are changed from \([0,1]\) to \([1,\infty)\). Finally, a generalization for higher dimensions is proved, which reveals an interesting link of those integrals to Mellin’s transform.