We study the $$\bar{\partial }$$ -equation first in Stein manifold then in complete Kähler manifolds. The aim is to get $$L^{r}$$ and Sobolev estimates on solutions with compact support. In the Stein case we get that for any (p, q)-form $$\omega$$ in $$L^{r}$$ with compact support and $$\bar{\partial }$$ -closed there is a $$(p,q-1)$$ -form u in $$W^{1,r}$$ with compact support and such that $$\bar{\partial } u=\omega .$$ In the case of Kähler manifold, we prove and use estimates on solutions on Poisson equation with compact support and the link with $$\bar{\partial }$$ equation is done by a classical theorem stating that the Hodge Laplacian is twice the $$\bar{\partial }$$ (or Kohn) Laplacian in a Kähler manifold. This uses and improves, in special cases, our result on Andreotti–Grauert-type theorem.

We study the $$\bar{\partial }$$ ∂ ¯ -equation first in Stein manifold then in complete Kähler manifolds. The aim is to get $$L^{r}$$ L r and Sobolev estimates on solutions with compact support. In the Stein case we get that for any ( p ,  q )-form $$\omega$$ ω in $$L^{r}$$ L r with compact support and $$\bar{\partial }$$ ∂ ¯ -closed there is a $$(p,q-1)$$ ( p , q - 1 ) -form u in $$W^{1,r}$$ W 1 , r with compact support and such that $$\bar{\partial } u=\omega .$$ ∂ ¯ u = ω . In the case of Kähler manifold, we prove and use estimates on solutions on Poisson equation with compact support and the link with $$\bar{\partial }$$ ∂ ¯ equation is done by a classical theorem stating that the Hodge Laplacian is twice the $$\bar{\partial }$$ ∂ ¯ (or Kohn) Laplacian in a Kähler manifold. This uses and improves, in special cases, our result on Andreotti–Grauert-type theorem.

We prove Sobolev embedding Theorems with weights for vector bundles in a complete Riemannian manifold. We also get general Gaffney’s inequality with weights. As a consequence, under a “weak bounded geometry” hypothesis, we improve classical Sobolev embedding Theorems for vector bundles in a complete Riemannian manifold. We also improve known results on Gaffney’s inequality in a complete Riemannian manifold.

We study the heat equation $\frac{\partial u}{\partial t}-\Delta u=0,\ u(x,0)=\omega (x),$ where $\Delta :=dd^{*}+d^{*}d$ is the Hodge laplacian and $u(\cdot ,t)$ and $\omega $ are $p$-differential forms in the complete Riemannian manifold $(M,g).$ Under weak bounded geometrical assumptions we get estimates on its semigroup of the form: acting on $p$-forms with $p\geq 1$ and $k\geq 0$: $\displaystyle \forall t\geq 1,\ {\left\Vert{\nabla ^{k}e^{-t\Delta_{p}}}\right\Vert}_{L^{r}(M)-L^{r}(M)}\leq c(n,r,k).$ Acting on functions, i.e. with $p=0,$ we get a better result: $\displaystyle \forall k\geq 1,\ \forall t\geq 1,\ {\left\Vert{\nabla ^{k}e^{-t\Delta }}\right\Vert}_{L^{r}(M)-L^{r}(M)}\leq c(n,r,k)t^{-1/2}.$

We use a special version of the Corona Theorem in several variables, valid when all but one of the data functions are smooth, to generalize to the polydisc and to the ball results obtained by El Fallah, Kellay and Seip about cyclicity of non vanishing bounded holomorphic functions in large enough Banach spaces of analytic functions determined either by weighted sums of powers of Taylor coefficients or by radially weighted integrals of powers of the modulus of the function.

We study Sobolev estimates for the solutions of parabolic equations acting on a vector bundle, in a complete, compact or non compact, riemannian manifold $M.$ The idea is to introduce geometric weights on $M.$ We get global Sobolev estimates with these weights. As applications, we find and improve "classical results", i.e. results without weights, by use of a Theorem by Hebey and Herzlich. As an example we get Sobolev estimates for the solutions of the heat equation on $p$-forms when the manifold has "weak bounded geometry " of order $1$.

By a theorem of Andreotti and Grauert if $\omega $ is a $(p,q)$ current, $q < n,$ in a Stein manifold $\displaystyle \Omega ,\ \bar \partial $ closed and with compact support, then there is a solution $u$ to $\bar \partial u=\omega $ still with compact support in $\displaystyle \Omega .$ The main result of this work is to show that if moreover $\displaystyle \omega \in L^{r}(m),$ where $m$ is a suitable Lebesgue measure on the Stein manifold, then we have a solution $u$ with compact support {\sl and} in $L^{s}(m),\ \frac{1}{s}=\frac{1}{r}-\frac{1}{2(n+1)}.$ We prove it by estimates in $L^{r}$ spaces with weights.

Our aim is to correct the proofs of Lemma 5.2 and Lemma 5.3

Let S be a sequence of points in Ω , where Ω is the unit ball or the unit polydisc in ℂ^n. Denote H^p(Ω) the Hardy space of Ω . Suppose that S is H^p interpolating with p≥ 2. Then S has the bounded linear extension property. The same is true for the Bergman spaces of the ball by use of the "Subordination Lemma". The point of view used here is the vectorial one: Hilbertian and Besselian basis.

Let S be a sequence of points in 𝔻^n. Suppose that S is H^p interpolating. Then we prove that the sequence S is Carleson, provided that p>2. We also give a sufficient condition, in terms of dual boundedness and Carleson measure, for S to be an H^p interpolating sequence.

We present two fast constructions of weak*-copies of l(infinity) in H-infinity, and show that such copies are necessarily weak*-complemented. Moreover, via a Paley-Wiener type of stability theorem for bases, a connection can be made in some cases between the two types of construction, via interpolating sequences (in fact these are at the basis of the second construction). Our approach has natural generalizations where H-infinity is replaced by an arbitrary dual space and l(infinity) by l(p) (1 <= p <= infinity), relying on the notions of generalized interpolating sequence and bounded linear extension. An old (very simple but unpublished so far) construction of bases which are Besselian but not Hilbertian finds a natural place in this development.

We study the $\bar \partial $-equation in complete K\ahler manifolds. The aim is to get $L^{r}$ and Sobolev estimates on solutions with compact support. We prove and use estimates on solutions on Poisson equation with compact support and the link with $\bar\partial $ equation is done by a classical theorem stating that the Hodge laplacian is twice the $\bar \partial $ (or Kohn) Laplacian in a K\ahler manifold. This uses and improves, in special cases, our result on Andreotti-Grauert type theorem.

Our aim is to correct the proofs of Lemma 5.2 and Lemma 5.3

Our aim is to correct the proofs of Lemma 5.2 and Lemma 5.3

We introduce the Local Increasing Regularity Method (LIRM) which allows us to get from local a priori estimates, on solutions u of a linear equation \(\displaystyle Du=\omega ,\) global ones. As an application we shall prove that if D is an elliptic linear differential operator of order m with \({\mathcal {C}}^{\infty }\) coefficients operating on the sections of a complex vector bundle \(\displaystyle G:=(H,\pi ,M)\) over a compact Riemannian manifold M without boundary and \(\omega \in L^{r}_{G}(M)\cap (\mathrm {k}\mathrm {e}\mathrm {r}D^{*})^{\perp },\) then there is a \(u\in W^{m,r}_{G}(M)\) such that \(Du=\omega \) on M. Next we investigate the case of a compact manifold with boundary by using the “Riemannian double manifold.” In the last sections we study the more delicate case of a complete but non-compact Riemannian manifold by the use of adapted weights.

In this work we study Hardy Sobolev spaces in the ball of $C^n$ with respect to interpolating sequences and Carleson measures. We compare them with the classical Hardy spaces of the ball and we stress analogies and differences.

Let A be a uniform algebra on the compact space X and σ a probability measure on X . We define the Hardy spaces H(σ) and the H(σ) interpolating sequences S in the p-spectrum Mp of σ. We prove, under some structural hypotheses on σ that ”Carleson type” conditions on S imply that S is interpolating with a linear extension operator in H(σ), s < p provided that either p = ∞ or p ≤ 2. This gives new results on interpolating sequences for Hardy spaces of the ball and the polydisc. In particular in the case of the unit ball of C we get that if there is a sequence {ρa}a∈S bounded in H(B) such that ∀a, b ∈ S, ρa(b) = δab, then S is H (B)-interpolating with a linear extension operator for any 1 ≤ p < ∞.

We generalize to intersection of strictly c-convex domains in Stein manifolds, Lr-Ls and Lipschitz estimates for the solutions of the equation obtained by Ma and Vassiliadou for domains in Cn. For this we use a Docquier-Grauert holomorphic retraction plus the rising steps method. This gives results in the case of intersection of domains with low regularity, C3, for their boundary.