On the Cauchy problem for convolution equations

"On the Cauchy problem for convolution equations." Colloquium Mathematicae 133.1 (2013): 115-132. <http://eudml.org/doc/286490>.

@article{Unknown2013,
abstract = {We consider one-parameter (C₀)-semigroups of operators in the space $^\{\prime \}(ℝⁿ;ℂ^\{m\})$ with infinitesimal generator of the form $(G*)|_\{^\{\prime \}(ℝⁿ;ℂ^\{m\})\}$ where G is an $M_\{m×m\}$-valued rapidly decreasing distribution on ℝⁿ. It is proved that the Petrovskiĭ condition for forward evolution ensures not only the existence and uniqueness of the above semigroup but also its nice behaviour after restriction to whichever of the function spaces $(ℝⁿ;ℂ^\{m\})$, $_\{L^\{p\}\}(ℝⁿ;ℂ^\{m\})$, p ∈ [1,∞], $(_\{a\})(ℝⁿ;ℂ^\{m\})$, a ∈ ]0,∞[, or the spaces $^\{\prime \}_\{L^\{q\}\}(ℝⁿ;ℂ^\{m\})$, q ∈ ]1,∞], of bounded distributions.},
journal = {Colloquium Mathematicae},
keywords = {one-parameter convolution semigroup; rapidly decreasing distribution; slowly increasing function; convolution algebra},
language = {eng},
number = {1},
pages = {115-132},
title = {On the Cauchy problem for convolution equations},
url = {http://eudml.org/doc/286490},
volume = {133},
year = {2013},
}

TY - JOUR
TI - On the Cauchy problem for convolution equations
JO - Colloquium Mathematicae
PY - 2013
VL - 133
IS - 1
SP - 115
EP - 132
AB - We consider one-parameter (C₀)-semigroups of operators in the space $^{\prime }(ℝⁿ;ℂ^{m})$ with infinitesimal generator of the form $(G*)|_{^{\prime }(ℝⁿ;ℂ^{m})}$ where G is an $M_{m×m}$-valued rapidly decreasing distribution on ℝⁿ. It is proved that the Petrovskiĭ condition for forward evolution ensures not only the existence and uniqueness of the above semigroup but also its nice behaviour after restriction to whichever of the function spaces $(ℝⁿ;ℂ^{m})$, $_{L^{p}}(ℝⁿ;ℂ^{m})$, p ∈ [1,∞], $(_{a})(ℝⁿ;ℂ^{m})$, a ∈ ]0,∞[, or the spaces $^{\prime }_{L^{q}}(ℝⁿ;ℂ^{m})$, q ∈ ]1,∞], of bounded distributions.
LA - eng
KW - one-parameter convolution semigroup; rapidly decreasing distribution; slowly increasing function; convolution algebra
UR - http://eudml.org/doc/286490
ER -