This paper is a survey of recent developments in two weight testing theory arising in joint projects with Tuomas Hyötnen, Kangwei Li, Chun-Yen Shen, Ignacio Uriarte-Tuero, Robert Rahm, and Brett Wick. An hour lecture on this work was presented in a talk at the Alamo Symposium in San Antonio, Texas in June of 2019. Our focus is the half-century-old research area of two weight inequalities for Calderón-Zygmund and other classical operators \(T\).

We extend results given by Zó and Cuenya in 2007 about a general approach to problems of best vector-valued approximation on small regions from a finite dimensional subspace of polynomials of some degree. This approach is called best local approximation. We consider a weighted local approximation of a vector-valued function on the origin and a weighted best local approximation of a real-valued function on several points, similar to classical problems in best local approximation with balanced neighborhood.

An integral is defined on the plane that includes the Henstock-Kurzweil and Lebesgue integrals (with respect to Lebesgue measure). A space of primitives is taken as the set of continuous real-valued functions \(F(x,y)\) defined on the extended real plane \([-\infty,\infty]^2\) that vanish when \(x\) or \(y\) is \(-\infty\). With usual pointwise operations this is a Banach space under the uniform norm. The integrable functions and distributions (generalised functions) are those that are the distributional derivative \(\partial^2/(\partial x\partial y)\) of this space of primitives. If \(f=\partial^2/(\partial x\partial y) F\) then the integral over interval \([a,b]\times [c,d] \subseteq[-\infty,\infty]^2\) is \(\int_a^b\int_c^d f=F(a,c)+F(b,d)-F(a,d)-F(b,c)\) and \(\int_{-\infty}^\infty \int_{-\infty}^\infty f=F(\infty,\infty)\). The definition then builds in the fundamental theorem of calculus. The Alexiewicz norm is \({\lVert f\rVert}={\lVert F\rVert}_\infty\) where \(F\) is the unique primitive of \(f\). The space of integrable distributions is then a separable Banach space isometrically isomorphic to the space of primitives. The space of integrable distributions is the completion of both \(L^1\) and the space of Henstock-Kurzweil integrable functions. The Banach lattice and Banach algebra structures of the continuous functions in \({\lVert \cdot\rVert}_\infty\) are also inherited by the integrable distributions. It is shown that the dual space is the functions of bounded Hardy-Krause variation. Various tools that make these integrals useful in applications are proved: integration by parts, Hölder’s inequality, second mean value theorem, Fubini’s theorem, a convergence theorem, change of variables, convolution. The changes necessary to define the integral in \({\mathbb R}^n\) are sketched out.

We provide an example of two \(2\)-periodic everywhere differentiable functions \(f,g\colon\mathbb{R}\to\mathbb{R}\) whose convolution \(f*g\) fails to be differentiable at every point of some perfect (thus, uncountable) set \(P\subset\mathbb{R}\). This shows that the convolution operator can actually destroy the differentiability of these maps, rather than introducing additional smoothness (as it is usually the case). New directions and open problems are also posed.

The boundedness of the weighted iterated Hardy-type operators \(T_{u,b}\) and \(T_{u,b}^*\) involving suprema from weighted Lebesgue space \(L_p(v)\) into weighted Cesàro function spaces \({\operatorname{Ces}}_{q}(w,a)\) are characterized. These results allow us to obtain the characterization of the boundedness of the supremal operator \(R_u\) from \(L^p(v)\) into \({\operatorname{Ces}}_{q}(w,a)\) on the cone of monotone non-increasing functions. For the convenience of the reader, we formulate the statement on the boundedness of the weighted Hardy operator \(P_{u,b }\) from \(L^p(v)\) into \({\operatorname{Ces}}_{q}(w,a)\) on the cone of monotone non-increasing functions. Under additional condition on \(u\) and \(b\), we are able to characterize the boundedness of weighted iterated Hardy-type operator \(T_{u,b}\) involving suprema from \(L^p(v)\) into \({\operatorname{Ces}}_q(w,a)\) on the cone of monotone non-increasing functions. At the end of the paper, as an application of obtained results, we calculate the norm of the fractional maximal function \(M_{\gamma}\) from \(\Lambda^p(v)\) into \(\Gamma^q(w)\).

Let \(A\) be a zero measure dense \(G_{\delta }\) subset of \(I=[0,1]\), with \(% \mathcal{M}\) the set of measurable self-maps of \(I\). There exists a residual set \(\mathcal{R}\subset \mathcal{M}\) such that for each \(f\) in \(\mathcal{R}\), the range of \(f\) is contained in \(A\), and the function \(f\) is one-to-one. Moreover, there exists \(h:I\rightarrow I\), a Baire-2 function, such that \(f(x)=h(x)\) \(a.e.\), and for any \(x\in I\), the trajectory \(\tau (x,h)\) is \(\infty \)-adic, so that the \(\omega \)-limit set \(\omega (x,h)\) is a Cantor set. Since the range of \(f\) is contained in \(A\), it follows that for any \(x\) in \(I\), there exists \(y\) in \(A\) such that the trajectory \(\tau (f(x),f)=\tau (y,f)\subset A\). Speaking loosely, the dynamical structures of \(f\) are completely determined by its behavior on the set \(A\).

Let \(K\) be a compact convex subset of \(\mathbb{R}^n\) whose Lebesgue measure is positive and which includes the origin of the coordinate system. In this paper, the density of the thinnest covering of \(\mathbb{R}^n\) by translates of \(K\) will be transformed into another form that seems easier to use. Finally, in the discussion section of this paper, we will look at the applications of the main results.

We show the Hyers-Ulam stability for points in the extended versions of Flett and Sahoo-Riedel theorems.

The coordinate functions, \(f\) and \(g\), of a space filling curve are continuous functions from \([0,1]\) to \([0,1]\) so that \(F(t)=(f(t),g(t))\) maps \([0,1]\) onto the unit square. In the paper Coordinate Functions of Space Filling Curves written by Foran, several necessary conditions for a continuous function \(f\) are given for there to be a continuous \(g\) so that \(F(t)=(f(t),g(t))\) maps \([0,1]\) onto \([0,1]^2\). In this paper, we define a new condition for \(f\) that is both necessary and sufficient to assure that \(f\) has a matching coordinate function \(g\) such that \(F(t)=(f(t),g(t))\) fills the square.

For \(0 \lt q \le p\) we take \(L^\beta\)-averages over the Hölder inequality between the \(l^q\)-norm and the \(l^p\)-norm in \(\mathbb{R}^n\). We obtain precise limits as \(n\to\infty\) for the \(l^p\)-unit ball and, in case \(p\ge1\), also for the \(l^p\)-unit sphere, which coincide and are independent of \(\beta>0\). These are consequences of more general results on the asymptotic behavior of corresponding integrals over balls and spheres of certain bounded measurable functions.

The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also introduce the concept of Banach statistical limit functional and the notion of GAS convergence mainly depends on the existence of Banach statistical limit functional. We prove the existence of Banach statistical limit functional. Then we have shown the existence of a GAS convergent sequence, which is neither statistical convergent nor almost convergent. Also, some topological properties of the space of all GAS convergent sequences are investigated.

We show that a typical Besicovitch set \(B\) has intersections of measure zero with every line not contained in it. Moreover, every line in \(B\) intersects the union of all the other lines in \(B\) in a set of measure zero.

An \(n\)-dimensional normed space is said to be symmetric if there exists a basis \(v_1, \dots, v_n\) such that \[ \left\| \sum_{i} |c_i| v_i \right\| = \left\| \sum_i c_{\sigma(i) } v_i \right\| \] for all \(c_1, \dots, c_n \in \mathbb R\) and for every \(\sigma \in \text{Perm}(n)\), the set of all permutations on \(n\) elements. In this paper we consider the existence of 2-dimensional symmetric subspaces of \(L^p[-1,1]\), spanned by \(t^k_e\) and \(t^k_o\) for even and odd integers \(k_e\) and \(k_o\). As the question of symmetry is basis-specific, we organize our consideration by (so-called) diagonal, triangular and symmetric bases. We find that, outside the \(p=4\) and \(p=2\) cases, symmetry is not possible for the bases considered.

A classical theorem of Riemann and Lebesgue says that a bounded function defined on a compact interval is Riemann integrable if and only if it is continuous almost everywhere. In this note, we generalize their result and show that the difference between the upper and lower Riemann integrals of a not necessarily Riemann integrable function equals the upper Riemann integral of its oscillation function.

This note corrects two misleading typographical errors in On the speed of convergence in the strong density theorem by P. Georgopoulos and C. Cryllakis, Real Anal. Exch. 44(1) 2019.