  The bulk of the molecular line emission in the double barred galaxy NGC4303
as observed in its CO(1-0) line with the OVRO mm-interferometer comes from two
straight gas lanes which run north-south along the leading sides of the
large-scale primary bar. Inside a radius of ~ 400 pc the molecular gas forms a
spiral pattern which, for the northern arm, can be traced to the nucleus.
Comparison of the OVRO and archival HST data with dynamical models of gas flow
in the inner kiloparsec of single- and double-barred galaxies shows that the
observed global properties of the molecular gas are in agreement with models
for the gas flow in a strong, large-scale bar, and the two-arm spiral structure
seen in CO in the inner kiloparsec can already be explained by a density wave
initiated by the potential of that bar. Only a weak correlation between the
molecular gas distribution and the extinction seen in the HST V-H map is found
in the inner 400 pc of NGC4303: The innermost part of one arm of the nuclear CO
spiral correlates with a weak dust filament in the color map, while the overall
dust distribution follows a ring or single-arm spiral pattern well correlated
with the UV continuum. This complicated nuclear geometry of the stellar and
gaseous components allows for two scenarios: (A) A self-gravitating m=1 mode is
present forming the spiral structure seen in the UV continuum. In this case the
gas kinematics would be unaffected by the small (~ 4'') inner bar. (B) The UV
continuum traces a complete ring which is heavily extincted north of the
nucleus. Such a ring forms in hydrodynamic models of double bars, but the
models cannot account for the UV emission observed on the leading side of the
inner bar. (abridged)
