Non-probabilistic proof of the A_2 theorem, and sharp weighted bounds for the q-variation of singular integrals

Any Calderon-Zygmund operator T is pointwise dominated by a convergent sum of positive dyadic operators. We give an elementary self-contained proof of this fact, which is simpler than the probabilistic arguments used for all previous results in this direction. Our argument also applies to the q-variation of certain Calderon-Zygmund operators, a stronger nonlinearity than the maximal truncations. As an application, we obtain new sharp weighted inequalities.Comment: 10 page

Remarks on functional calculus for perturbed first order Dirac operators

We make some remarks on earlier works on $R-$bisectoriality in $L^p$ of perturbed first order differential operators by Hyt\"onen, McIntosh and Portal. They have shown that this is equivalent to bounded holomorphic functional calculus in $L^p$ for $p$ in any open interval when suitable hypotheses are made. Hyt\"onen and McIntosh then showed that $R$-bisectoriality in $L^p$ at one value of $p$ can be extrapolated in a neighborhood of $p$. We give a different proof of this extrapolation and observe that the first proof has impact on the splitting of the space by the kernel and range.Comment: 11 page

Conical square function estimates in UMD Banach spaces and applications to H-infinity functional calculi

We study conical square function estimates for Banach-valued functions, and introduce a vector-valued analogue of the Coifman-Meyer-Stein tent spaces. Following recent work of Auscher-McIntosh-Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator (A) with certain off-diagonal bounds, such that (A) always has a bounded (H^{\infty})-functional calculus on these spaces. This provides a new way of proving functional calculus of (A) on the Bochner spaces (L^p(\R^n;X)) by checking appropriate conical square function estimates, and also a conical analogue of Bourgain's extension of the Littlewood-Paley theory to the UMD-valued context. Even when (X=\C), our approach gives refined (p)-dependent versions of known results.Comment: 28 pages; submitted for publicatio

Pointwise convergence of vector-valued Fourier series

We prove a vector-valued version of Carleson's theorem: Let Y=[X,H]_t be a complex interpolation space between a UMD space X and a Hilbert space H. For p\in(1,\infty) and f\in L^p(T;Y), the partial sums of the Fourier series of f converge to f pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form Y=[X,H]_t. In particular, we answer affirmatively a question of Rubio de Francia on the pointwise convergence of Fourier series of Schatten class valued functions.Comment: 26 page

Pointwise convergence of Walsh-Fourier series of vector-valued functions

We prove a version of Carleson’s Theorem in the Walsh model for vector-valued functions: For 1<p<∞, and a UMD space Y, the Walsh–Fourier series of f∈Lp(0,1;Y) converges pointwise, provided that Y is a complex interpolation space Y=[X,H]θ between another UMD space X and a Hilbert space H, for some θ∈(0,1). Apparently, all known examples of UMD spaces satisfy this condition.Peer reviewe

Complete measurements of quantum observables

We define a complete measurement of a quantum observable (POVM) as a measurement of the maximally refined version of the POVM. Complete measurements give information from the multiplicities of the measurement outcomes and can be viewed as state preparation procedures. We show that any POVM can be measured completely by using sequential measurements or maximally refinable instruments. Moreover, the ancillary space of a complete measurement can be chosen to be minimal.Comment: Based on talk given in CEQIP 2012 conferenc