Let $mathcal{M}$ be a semifinite von Neumann algebra. We equip the associated noncommutative $L_p$-spaces with their natural operator space structure introduced by Pisier via complex interpolation. On the other hand, for $1<p<infty$ let $$L_{p,p}(mat
hcal{M})=big(L_{infty}(mathcal{M}),,L_{1}(mathcal{M})big)_{frac1p,,p}$$ be equipped with the operator space structure via real interpolation as defined by the second named author ({em J. Funct. Anal}. 139 (1996), 500--539). We show that $L_{p,p}(mathcal{M})=L_{p}(mathcal{M})$ completely isomorphically if and only if $mathcal{M}$ is finite dimensional. This solves in the negative the three problems left open in the quoted work of the second author. We also show that for $1<p<infty$ and $1le qleinfty$ with $p eq q$ $$big(L_{infty}(mathcal{M};ell_q),,L_{1}(mathcal{M};ell_q)big)_{frac1p,,p}=L_p(mathcal{M}; ell_q)$$ with equivalent norms, i.e., at the Banach space level if and only if $mathcal{M}$ is isomorphic, as a Banach space, to a commutative von Neumann algebra. Our third result concerns the following inequality: $$ big|big(sum_ix_i^qbig)^{frac1q}big|_{L_p(mathcal{M})}lebig|big(sum_ix_i^rbig)^{frac1r}big|_{L_p(mathcal{M})} $$ for any finite sequence $(x_i)subset L_p^+(mathcal{M})$, where $0<r<q<infty$ and $0<pleinfty$. If $mathcal{M}$ is not isomorphic, as a Banach space, to a commutative von Meumann algebra, then this inequality holds if and only if $pge r$.
Let ${mathbb{P}_t}_{t>0}$ be the classical Poisson semigroup on $mathbb{R}^d$ and $G^{mathbb{P}}$ the associated Littlewood-Paley $g$-function operator: $$G^{mathbb{P}}(f)=Big(int_0^infty t|frac{partial}{partial t} mathbb{P}_t(f)|^2dtBig)^{frac12}.
$$ The classical Littlewood-Paley $g$-function inequality asserts that for any $1<p<infty$ there exist two positive constants $mathsf{L}^{mathbb{P}}_{t, p}$ and $mathsf{L}^{mathbb{P}}_{c, p}$ such that $$ big(mathsf{L}^{mathbb{P}}_{t, p}big)^{-1}big|fbig|_{p}le big|G^{mathbb{P}}(f)big|_{p} le mathsf{L}^{mathbb{P}}_{c,p}big|fbig|_{p},,quad fin L_p(mathbb{R}^d). $$ We determine the optimal orders of magnitude on $p$ of these constants as $pto1$ and $ptoinfty$. We also consider similar problems for more general test functions in place of the Poisson kernel. The corresponding problem on the Littlewood-Paley dyadic square function inequality is investigated too. Let $Delta$ be the partition of $mathbb{R}^d$ into dyadic rectangles and $S_R$ the partial sum operator associated to $R$. The dyadic Littlewood-Paley square function of $f$ is $$S^Delta(f)=Big(sum_{RinDelta} |S_R(f)|^2Big)^{frac12}.$$ For $1<p<infty$ there exist two positive constants $mathsf{L}^{Delta}_{c,p, d}$ and $ mathsf{L}^{Delta}_{t,p, d}$ such that $$ big(mathsf{L}^{Delta}_{t,p, d}big)^{-1}big|fbig|_{p}le big|S^Delta(f)big|_{p}le mathsf{L}^{Delta}_{c,p, d}big|fbig|_{p},quad fin L_p(mathbb{R}^d). $$ We show that $$mathsf{L}^{Delta}_{t,p, d}approx_d (mathsf{L}^{Delta}_{t,p, 1})^d;text{ and }; mathsf{L}^{Delta}_{c,p, d}approx_d (mathsf{L}^{Delta}_{c,p, 1})^d.$$ All the previous results can be equally formulated for the $d$-torus $mathbb{T}^d$. We prove a de Leeuw type transference principle in the vector-valued setting.
We study vector-valued Littlewood-Paley-Stein theory for semigroups of regular contractions ${T_t}_{t>0}$ on $L_p(Omega)$ for a fixed $1<p<infty$. We prove that if a Banach space $X$ is of martingale cotype $q$, then there is a constant $C$ such that
$$ left|left(int_0^inftybig|tfrac{partial}{partial t}P_t (f)big|_X^q,frac{dt}tright)^{frac1q}right|_{L_p(Omega)}le C, big|fbig|_{L_p(Omega; X)},, quadforall, fin L_p(Omega; X),$$ where ${P_t}_{t>0}$ is the Poisson semigroup subordinated to ${T_t}_{t>0}$. Let $mathsf{L}^P_{c, q, p}(X)$ be the least constant $C$, and let $mathsf{M}_{c, q}(X)$ be the martingale cotype $q$ constant of $X$. We show $$mathsf{L}^{P}_{c,q, p}(X)lesssim maxbig(p^{frac1{q}},, pbig) mathsf{M}_{c,q}(X).$$ Moreover, the order $maxbig(p^{frac1{q}},, pbig)$ is optimal as $pto1$ and $ptoinfty$. If $X$ is of martingale type $q$, the reverse inequality holds. If additionally ${T_t}_{t>0}$ is analytic on $L_p(Omega; X)$, the semigroup ${P_t}_{t>0}$ in these results can be replaced by ${T_t}_{t>0}$ itself. Our new approach is built on holomorphic functional calculus. Compared with all the previous, the new one is more powerful in several aspects: a) it permits us to go much further beyond the setting of symmetric submarkovian semigroups; b) it yields the optimal orders of growth on $p$ for most of the relevant constants; c) it gives new insights into the scalar case for which our orders of the best constants in the classical Littlewood-Paley-Stein inequalities for symmetric submarkovian semigroups are better than the previous by Stein. In particular, we resolve a problem of Naor and Young on the optimal order of the best constant in the above inequality when $X$ is of martingale cotype $q$ and ${P_t}_{t>0}$ is the classical Poisson and heat semigroups on $mathbb{R}^d$.
We study Fourier multipliers on free group $mathbb{F}_infty$ associated with the first segment of the reduced words, and prove that they are completely bounded on the noncommutative $L^p$ spaces $L^p(hat{mathbb{F}}_infty)$ iff their restriction on $L
^p(hat{mathbb{F}}_1)=L^p(mathbb{T})$ are completely bounded. As a consequence, every classical Mikhlin multiplier extends to a $L^p$ Fourier multiplier on free groups for all $1<p<infty$.
We consider the reduction of problems on general noncommutative $L_p$-spaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is a unpublished result of the first named author which approximates any noncom
mutative $L_p$-space by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products by a locally compact abelian group or to their noncommutative $L_p$-spaces. We present applications of these results to the theory of noncommutative martingale inequalities by reducing most recent general noncommutative martingale/ergodic inequalities to those in the tracial case.
This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $mathbb T$, respectively. For $1< q<inft
y$ and a Banach space $B$ we prove that there exists a positive constant $c$ such that $$sup_{z_0in D}int_{D}(1-|z|)^{q-1}| abla f(z)|^q P_{z_0}(z) dA(z) le c^qsup_{z_0in D}int_{T}|f(z)-f(z_0)|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ iff $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=frac{1-|z_0|^2}{|1-bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.
This paper is withdrawn. We found a mistake in Lemma 4.1
We show norm estimates for the sum of independent random variables in noncommutative $L_p$-spaces for $1<p<infty$ following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. Among applications,
we derive an equivalence for the $p$-norm of the singular values of a random matrix with independent entries, and characterize those symmetric subspaces and unitary ideals which can be realized as subspaces of a noncommutative $L_p$ for $2<p<infty$.
Let $A$ be a finite subdiagonal algebra in Arvesons sense. Let $H^p(A)$ be the associated noncommutative Hardy spaces, $0<ple8$. We extend to the case of all positive indices most recent results about these spaces, which include notably the Riesz, Sz
ego and inner-outer type factorizations. One new tool of the paper is the contractivity of the underlying conditional expectation on $H^p(A)$ for $p<1$.
