arXiv subjects

Ramesh Manna

Publications and source records attributed to Ramesh Manna.

22 records · Page 2Linked to original sources

arXiv2018

Space like strong unique continuation for sublinear parabolic equations

In this paper, we establish space like strong unique continuation property (sucp) for uniformly parabolic sublinear equations under appropriate structural assumptions. Our main result Theorem 1.1 constitutes the parabolic counterpart of the strong unique continuation result recently established by Ruland in [Ru] for analogous elliptic sublinear equations. Similar to that in [Ru], this is accomplished via a new $L^{2}-L^{2}$ type Carleman estimate for a class of sublinear parabolic operators.

math.AP↗

arXiv2018

Momentum ray transforms

The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines with the weight $t^k$: $ (I^k\!f)(x,ξ)=\int_{-\infty}^\infty t^k\langle f(x+tξ),ξ^m\rangle\,dt. $ In particular, the ray transform $I=I^0$ was studied by several authors since it had many tomographic applications. We present an algorithm for recovering $f$ from the data $(I^0\!f,I^1\!f,\dots, I^m\!f)$. In the cases of $m=1$ and $m=2$, we derive the Reshetnyak formula that expresses $\|f\|_{H^s_t({\mathbb{R}}^n)}$ through some norm of $(I^0\!f,I^1\!f,\dots, I^m\!f)$. The $H^{s}_{t}$-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.

math.AP↗

arXiv2018

Maximal functions associated to flat plane curves with Mitigating factors

We study the boundedness problem for maximal operators $\mathbb{M}_σ$ associated to flat plane curves with Mitigating factors, defined by $$\mathbb{M}_σf(x) \, := \, \sup_{1 \leq t \leq 2} \left|\int_{0}^{1} f(x-tΓ(s)) \, (κ(s))^σ \, ds\right|,$$ where $κ(s)$ denotes the curvature of the curve $Γ(s)=(s, g(s)+1), ~g(s) \in C^5[0,1]$ in $\mathbb{R}^2$. Let $\triangle$ be the closed triangle with vertices $P=(\frac{2}{5}, \frac{1}{5}), ~ Q=(\frac{1}{2}, \frac{1}{2}), ~ R=(0, 0).$ In this paper, we prove that for $ (\frac{1}{p}, \frac{1}{q}) \in \left[(\frac{1}{p}, \frac{1}{q}) :(\frac{1}{p}, \frac{1}{q}) \in \triangle \setminus \{P, Q\} \right] \cap \left[(\frac{1}{p}, \frac{1}{q}) :q > max\{σ^{-1},2\} \right]$, there is a constant $B$ such that $ \|\mathbb{M}f\|_{L^q(\mathbb{R}^2)} \leq \, B \, \|f\|_{L^p(\mathbb{R}^2)}. $

math.CA↗

arXiv2016

Maximal averages associated to families of finite type surfaces

We study the boundedness problem for maximal operators $\mathcal{M}$ associated to averages along families of hypersurfaces $S$ of finite type in $\mathbb{R}^n.$ In this paper, we prove that if $S$ is a finite type hypersurface which is of finite type $k$ at $x_0 \in \mathbb{R}^n$, then the associated maximal operator is bounded on $L^p(\mathbb{R}^n)$ for $p>k.$ We shall also consider a variable coefficient version of maximal theorem and we obtain the same $L^p-$ boundedness result for $p>k.$ We also discuss the consequence of this result. In particular, we verify a conjecture by E. M. Stein and its generalization by A. Iosevich and E. Sawyer on the connection between the decay rate of the Fourier transform of the surface measure on $S$ and the $L^p-$ boundedness of the associated maximal operator $\mathcal{M}.$

math.CA↗